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ICME-13 Monographs

Patricio Herbst
Ui Hock Cheah
Philippe R. Richard
Keith Jones Editors

International
Perspectives on
the Teaching and
Learning of Geometry
in Secondary Schools
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

Patricio Herbst Ui Hock Cheah
•

Philippe R. Richard Keith Jones

•

Editors

International Perspectives
on the Teaching and Learning
of Geometry in Secondary
Schools

123
Editors
Patricio Herbst Philippe R. Richard
School of Education Département de didactique
University of Michigan Université de Montréal
Ann Arbor, MI Montreal, QC
USA Canada

Ui Hock Cheah Keith Jones

Institute of Teacher Education School of Education
Penang University of Southampton
Malaysia Southampton
UK

ISSN 2520-8322 ISSN 2520-8330 (electronic)

ICME-13 Monographs
ISBN 978-3-319-77475-6 ISBN 978-3-319-77476-3 (eBook)
https://doi.org/10.1007/978-3-319-77476-3
Library of Congress Control Number: 2018934401

© 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
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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
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Printed on acid-free paper

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The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents

1 International Perspectives on Secondary Geometry Education:

An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Patricio Herbst, Ui Hock Cheah, Keith Jones and Philippe R. Richard
2 Thinking About the Teaching of Geometry Through
the Lens of the Theory of Geometric Working Spaces . . . . . . . . . . 5
Alain Kuzniak
3 Epistemological Features of a Constructional Approach
to Regular 4-Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Stephan Berendonk and Marc Sauerwein
4 Opportunities for Reasoning and Proving in Geometry
in Secondary School Textbooks from Trinidad and Tobago . . . . . . 39
Andrew A. Hunte
5 Enacting Functions from Geometry to Algebra . . . . . . . . . . . . . . . . 59
Scott Steketee and Daniel Scher
6 Examining the Work of Teaching Geometry as a Subject-Specific
Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Patricio Herbst, Nicolas Boileau and Umut Gürsel
7 Differences in Self-reported Instructional Strategies Using
a Dynamic Geometry Approach that Impact Students’
Conjecturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Brittany Webre, Shawnda Smith and Gilbert Cuevas
8 Creating Profiles of Geometry Teachers’ Pedagogical
Content Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Agida G. Manizade and Dragana Martinovic
9 Symbiosis Between Subject Matter and Pedagogical
Knowledge in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Mohan Chinnappan, Bruce White and Sven Trenholm

v
vi Contents

10 Minding the Gap: A Comparison Between Pre-service

and Practicing High School Teachers’ Geometry
Teaching Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Shawnda Smith
11 Designing Instruction in Geometry: Using Lesson Study
to Improve Classroom Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Ui Hock Cheah
12 A Professional Development Experience in Geometry
for High School Teachers: Introducing Teachers
to Geometry Workspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
José Villella, Gema Fioriti, Rosa Ferragina, Leonardo Lupinacci,
Fernando Bifano, Susana Amman and Alejandra Almirón
13 Development of Spatial Ability: Results from the Research
Project GeodiKon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Guenter Maresch
14 Middle School Students’ Use of Property Knowledge
and Spatial Visualization in Reasoning About 2D Rotations . . . . . . 231
Michael T. Battista and Leah M. Frazee
15 Exploring Models of Secondary Geometry Achievement . . . . . . . . . 265
Sharon L. Senk, Denisse R. Thompson, Yi-Hsin Chen
and Kevin J. Voogt
16 Engaging Students with Non-routine Geometry Proof Tasks . . . . . . 283
Michelle Cirillo
17 Aspects of Spatial Thinking in Problem Solving: Focusing on
Viewpoints in Constructing Internal Representations . . . . . . . . . . . 301
Mitsue Arai
18 Playing with Geometry: An Educational Inquiry Game Activity . . . . 319
Yael Luz and Carlotta Soldano
19 The Use of Writing as a Metacognitive Tool in Geometry
Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Luz Graciela Orozco Vaca
20 Connectedness of Problems and Impasse Resolution
in the Solving Process in Geometry: A Major
Educational Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Philippe R. Richard, Michel Gagnon and Josep Maria Fortuny
21 Conclusion: Prospects for Developments and Research
in Secondary Geometry Education . . . . . . . . . . . . . . . . . . . . . . . . . 377
Patricio Herbst, Ui Hock Cheah, Keith Jones and Philippe R. Richard
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
Chapter 1
International Perspectives on Secondary
Geometry Education: An Introduction

Patricio Herbst, Ui Hock Cheah, Keith Jones and Philippe R. Richard

Abstract This chapter introduces the book by providing an orientation to the field
of research and practice in the teaching and learning of secondary geometry. The
editors describe the chapters in the book in terms of how they contribute to address
questions asked in the field, outlining different reasons why prospective readers
might want to look into specific chapters.

Keywords Curriculum
Thinking
Learning
Teaching
Teacher knowledge

This book is one of the outcomes of Topic Study Group 13 at the 13th International
Congress on Mathematical Education, which took place in Hamburg, Germany, in
the summer of 2016. Our Topic Study Group (TSG-13) concerned the teaching and
learning of secondary geometry and the chapters in this volume include revised
versions of most of the papers presented at the main meetings of the group. Also
included are a handful of the shorter papers associated with TSG-13 in the context
of short oral communications. In this brief introduction we orient the reader to these
papers by first providing an organizer of the focus of our study group.
The International Congress in Mathematics Education gathers researchers and
practitioners in mathematics education and pursues a goal of inclusiveness across
all sorts of boundaries. In particular, the boundaries between research and practice

P. Herbst (&)
University of Michigan, Ann Arbor, USA
e-mail: pgherbst@umich.edu
U. H. Cheah
Institut Pendidikan Guru Malaysia, Penang, Malaysia
e-mail: uhcrecsam1@gmail.com
K. Jones
University of Southampton, Southampton, UK
e-mail: d.k.jones@soton.ac.uk
P. R. Richard
Université de Montréal, Montréal, Canada
e-mail: philippe.r.richard@umontreal.ca

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

P. Herbst et al. (eds.), International Perspectives on the Teaching and Learning
of Geometry in Secondary Schools, ICME-13 Monographs,
https://doi.org/10.1007/978-3-319-77476-3_1
2 P. Herbst et al.

are often blurred in ICME and this surely applied to our Topic Study Group 13 in
ICME-13. Therefore, to orient the reader to the chapters in the book, it might be
useful to describe the territory or field of practice associated with the teaching and
learning of secondary geometry.
As we engage in such a description, we might benefit from using the metaphor of
map-making as a guiding principle. Borges’s short story On exactitude in science
uncovers the futility of expecting that a map be produced on a scale 1:1. Yet the
value of maps as containers of geographic knowledge and as resources for travelers
cannot be overemphasized, even if the existence of different kinds of projection
techniques reminds us that any map has limitations in what it affords its readers.
Different maps afford us different kinds of insight on the territory.
There is a constellation of practices that might be spotted as we look toward the
teaching and learning of geometry in secondary schools. At the center of this con-
stellation is the classroom practice of students and teacher transacting geometric
meanings. Near that center one can find the practice of textbook writing and materials
development for secondary geometry; one can also find the practice of preparing
teachers to teach secondary school geometry; and the individual practice of thinking
and problem solving that youngsters of secondary school age may engage in even
outside of school. But as we look closer, finer, relevant distinctions can be made.
The practice of teaching and learning geometry in classrooms admits of one set of
distinctions regarding the institutional location of those classrooms: American
secondary schools locate that practice in a single high school geometry course, while
geometry is integrated with other content areas in most other countries, and also
occurs outside of compulsory education, in other organized settings such as summer
camps. None of our papers inquires specifically on the institutional situatedness of
geometry instruction, though Kuzniak’s chapter recommends investigating whether
there is a place for the study of geometry in all educational systems, and uses a
contrast between work observed in Chile and in France as a way into his approach to
questioning the nature of geometric work. Other chapters present inquiries that seem
to rely on such situatedness. The chapter by Berendonk and Sauerwein, for example,
describes geometry experiences with novel content in the context of a summer
course for mathematically-inclined students, and the chapter by Herbst, Boileau, and
Gürsel examines how the instructional situations that are customary in the US high
school geometry course serve to frame a novel geometry task. Steeped into the
institutional location of the teaching and learning of geometry in high school in the
United States, Senk, Thompson, Chen, and Voogt examine outcomes of geometry
courses taught using the Geometry text from the University of Chicago School
Mathematics Program. Likewise Hunte’s chapter examines curricular variations
situated in the context of textbooks of different eras in Trinidad and Tobago.
Specific geometry content at stake in classroom instruction, as well as in teacher
development, textbook writing, and thinking and problem solving is discussed
implicitly or explicitly in all chapters. Several chapters focus on specific geometric
concepts: area of trapezoids (Manizade and Martinovic’s chapter), area of triangles
(Cheah’s chapter), properties of quadrilaterals (Herbst, Boileau, and Gürsel’s
chapter), polytopes (Berendonk and Sauerwein’s chapter), rotations (Battista and
1 International Perspectives on Secondary Geometry Education … 3

Frazee’s chapter), and connections to functions (Steketee and Scher’s chapter).

Specific geometric processes are also present as Hunte’s chapter deals with the
work of calculating, the chapter by Chinnappan, White and Trenholm includes
descriptions of the work of constructing, Luz and Soldano’s paper addresses the
work of conjecturing, and Cirillo’s paper deals with the work of proving.
The nature of and difficulties in students’ thinking, learning, achievement, and
problem solving in geometry are under consideration in several chapters. Across
these chapters there is attention to spatial thinking and to aspects of deductive rea-
soning from conjecturing to proving. Maresch’s chapter is focused on students’
spatial capabilities, Arai’s chapter deals with how students answer spatial orientation
tasks, and Battista and Frazee provide detailed descriptions of how students reason in
the context of rotation tasks. The chapter by Cirillo describes successful and
unsuccessful students’ thinking and collaboration in proof tasks. Similarly, Webre,
Smith, and Cuevas address the time and quality of students’ conjecturing in con-
nection with their engagement in discussions. And the chapter by Luz and Soldano
demonstrates how computer-based games engage students in conjecturing and fal-
sifying. Many of those processes are involved in the explorations proposed by Villella
and his collaborators. Senk and her colleagues map the variability in students’
achievement in a geometry test and look for ways to account for such variability.
The role of tools and resources in geometry instruction, thinking, materials
development, and teacher development is also quite apparent. The technological
mediation of materials development in geometry is eloquently illustrated by
Steketee and Scher in their chapter showing how dynamic geometry provides a
different access to the connections between functions and geometry. Technological
mediation of students’ thinking and learning is present in the chapter by Battista and
Frazee who illustrate the use of iDGi in eliciting students reasoning. Also dis-
cussing the mediation of students’ thinking, Luz and Soldano demonstrate how
games can be developed through dynamic geometry, internet communication, and
turn-taking. The role of Dynamic Geometry Software in teacher development is
discussed in the chapter by Villella and associates, while Webre and her colleagues
make comparable points in the case of classroom instruction. Richard, Gagnon, and
Fortuny add intelligent tutoring to dynamic geometry. This chapter’s focus is on
students’ blockage during geometric problem solving and how an intelligent tutor
can support students’ thinking. Along with Orozco’s chapter on the role of writing,
these last two help the book connect issues of mediation to metacognition.
Instruments for geometry instruction, thinking, materials development, or tea-
cher development need not be technological though. The chapter by Cheah
describes the use of the professional development practice called lesson study in the
design and planning of a lesson on area by a group of teachers. The chapter by
Herbst and his colleagues examines how a teacher made use of instructional situ-
ations of exploration, construction, and proof, which were available in her class, to
frame a novel geometry task on quadrilaterals as it was implemented in a geometry
course. The chapter by Chinnappan, White, and Trenholm describes the work of
teaching geometry in terms of its use of specialized and pedagogical content
knowledge. As regards the development of ways of assessing teacher knowledge
4 P. Herbst et al.

Manizade and Martinovic demonstrate how they use student work to elicit teachers’
responses that allow them to assess what they know about specific geometric topics.
In contrast, Smith uses the MKT-G test (Herbst & Kosko, 2014) to measure the
amount of mathematical knowledge for teaching geometry of practicing and pre-
service teachers across the domains hypothesized by Ball, Thames, and Phelps
(2008). Additionally, Smith uses a questionnaire to access self-reported pedagogical
practices of her participants. Also, the chapter by Villella and his colleagues from
Grupo CEDE describes how teachers’ knowledge of geometry can be developed
through experiences framed using ideas from the theory of geometric working
spaces introduced earlier in Kuzniak’s chapter.
As the chapters address those practices, they do so from multiple perspectives
that cover the range between practitioner and researcher. The chapters by
Berendonk and Sauerwein and by Steketee and Scher illustrate the work of
developing curriculum materials for the teaching of geometry. The development of
assessments for teachers is showcased in the paper by Manizade and Martinovic,
while the development of games for students is showcased in the paper by Luz and
Soldano. The chapter by Cheah illustrates the work of engaging teachers in pro-
fessional development using lesson study, while the chapter by Villella et al.
describes activities used in other professional development activities. The chapters
by Maresch, by Senk et al., and by Smith are based, at least in part, on the use of
tests. The observation of actual classroom interaction is present in a number of
papers including, in particular, Chinnappan et al.’s chapter and Herbst et al.’s
chapter. We come back in the conclusion to some methodological aspects of the
work presented.
The various ways in which we map the practices of teaching and learning ge-
ometry in secondary school highlight many connections and distinctions among the
chapters in the book. Surely more can be found through reading and with such
purpose we invite the reader to dig in. The book represents a collaborative effort
among editors in four different countries (Canada, Malaysia, the United Kingdom,
and the United States) working alongside 40 authors, affiliated with 25 different
institutions from 14 different countries. These authors put together 21 chapters. In
such representation of diversity, this book not only represents diverse perspectives on
the practice of teaching and learning geometry in secondary schools, but also rep-
resents the diversity among the individuals who attended ICME-13. May this diverse
offering of ideas inspire the reader to become a contributor to ICME in the future.

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it
special? Journal of Teacher Education, 59(5), 389–407.
Herbst, P., & Kosko, K. W. (2014). Mathematical knowledge for teaching and its specificity to
high school geometry instruction. In J. Lo, K. R. Leatham, & L. R. Van Zoest (Eds.), Research
trends in mathematics teacher education (pp. 23–46). New York: Springer.
Chapter 2
Thinking About the Teaching
of Geometry Through the Lens
of the Theory of Geometric Working
Spaces

Alain Kuzniak

Abstract In this communication, I argue that shared theoretical frameworks and

specific topics need to be developed in international research in geometry education
to move forward. My purpose is supported both by my experience as chair and
participant in different international conferences (CERME, ICME), and also by a
research program on Geometric Working Spaces and geometric paradigms. I show
how this framework allows thinking about the nature of geometric work in various
educational contexts.

Keywords Construction Discursive dimension Geometric work




Geometric paradigms Geometric working space Instrumental dimension




Proof Register of representation Reasoning Semiotic dimension
Visualization

2.1 Introduction

The purpose of this essay is not to give a general and critical overview of research
done in the domain of geometry education. First, this type of survey already exists
(e.g., the recent and very interesting ICME-13 survey team report, Sinclair et al.,
2016), and secondly, because given the extension of this field, such surveys are
generally partial and, sometimes, even biased. Indeed, geometry is taught from
kindergarten to university in many countries, and students engage with it in
very different ways, eventually depending on their professional orientation

A. Kuzniak (&)
Laboratoire de Didactique André Revuz, Paris Diderot University, Paris, France
e-mail: kuzniak@math.univ-paris-diderot.fr

© Springer International Publishing AG, part of Springer Nature 2018 5

P. Herbst et al. (eds.), International Perspectives on the Teaching and Learning
of Geometry in Secondary Schools, ICME-13 Monographs,
https://doi.org/10.1007/978-3-319-77476-3_2
6 A. Kuzniak

(e.g., architects, craft-persons, engineers, mathematics researchers). Geometry is

also a main topic in the preparation of primary and secondary school teachers.
Rather, what I want to do in this contribution is to formulate some ideas based on
my experience as researcher involved both in the CERME geometry working
group, which I was lucky to participate in or chair several times, and in the
development of an original model designed for the analysis of issues related to the
teaching of geometry, but also for comparative studies of this teaching in various
countries.
In one of his rare articles on the teaching of geometry, Brousseau (1987) insists
on the need of finding a substitute for the “natural” epistemological vigilance one
would expect from mathematicians but which is missing on account of the
extinction of any mathematical research on elementary geometry: This substitute
would enable the field to avoid the uncontrolled didactification of geometry that
Brousseau finds in teachers’ practices. Brousseau stresses the essential relationship
between epistemology and didactics in the teaching of geometry. In my view, this
search for a source of vigilance should pass through well-identified research
themes, and be based on development of shared theoretical frameworks in geometry
education even if they can be diverse to be adapted in a variety of contexts.
During the symposium honoring Artigue in Paris in 2012, Boero (2016) drew
the audience’s attention to the fact that the role of researchers in mathematics
education depends on strong cultural and institutional components that vary from
one country to another. In his country, Italy, researchers in the domain have to be
active in two opposite directions: In developing innovation and textbooks with an
immediate impact on the country’s school life, and at the same time, in developing a
research field which can be independent of immediate applications. In all countries,
in some form, researchers should be involved to influence education in the country
in which they live. But at the same time and independently of any political pressure,
they should also evaluate and compare existing teaching activities by researching
their effects on the actual mathematical development of students faced with such set
of tasks. In addition, research must, as far as possible, highlight and explore
invariant parameters that may exist in different contexts. Furthermore,
well-accepted findings in didactics of geometry should be known and taken into
account by researchers to ensure progress in the domain. Even when this is far from
easy, the field of research on geometry education would benefit from being struc-
tured around theoretical frameworks and specific research themes to stop being
always an emergent scientific domain. Supported by the model of Geometric
Working Spaces (GWS) and the related notion of geometric paradigms, I develop a
possible approach in this direction. Naturally, the GWS model is only used as an
example to show the possible interest of theoretical approaches in the domain.
Indeed, a diversity of theoretical approaches is needed to address the wide variety of
issues in such an extended field as geometry education.
2 Thinking About the Teaching of Geometry Through the Lens … 7

2.2 Travel in a Changing Territory Constantly

in Reconstruction

The difficulty of developing research and a common theoretical framework in

geometry education comes first from its chaotic evolution over the last decades. In
the early sixties, the French mathematician Dieudonné became widely known in the
education field by his famous cry “Euclid must go!” At the time, he wanted to
denounce a mathematical education ossified around notions that he considered
outdated and, in particular, what was called the geometry of the triangle. He did not
wish to destroy the teaching of geometry but rather to promote a consistent teaching
of this domain, based on more recent mathematical research and, particularly,
focusing on algebraic structures. According to Dieudonné, students should enter
directly into the most powerful mathematics without any long detours through
concepts and techniques that he considered obsolete. This questioning of traditional
geometry education initiated a series of reforms and counter-reforms. While some
of those reforms sought to bring school geometry closer to the geometry of
mathematicians, others have been sought to avoid learning difficulties that students
had faced. The teaching of geometry has become more and more utilitarian over
time, as exemplified and guided by the PISA expectations.
Furthermore, the teaching of geometry is marked by a great variability among
curricula across countries, which makes difficult the consistent networking of
researchers on specific topics. This variability can be illustrated by the place that
geometric transformations have had since the early seventies to the present in the
French curriculum.1 In the 1970s, heavily influenced by the mathématiques mod-
ernes (i.e., the new Math), geometric transformations such as translations and
similarities were used to separate affine and Euclidean properties. Then in the
1980s, transformations were studied in close relation with linear algebra and ana-
lytic work in two and three dimensions. There was then also important work on
how symmetries generate isometric transformations. In the 1990s, the work became
more geometric and transformations were limited to the plane and to explore
configurations like regular polygons, as transformations were implicitly associated
with the dihedral groups of polygons and the group of similarities associated with
complex numbers was the culmination of that mathematical journey. In the 2000s,
the importance of transformations decreased again with the disappearance of
dilations and similarities. As of 2008, translations and symmetries were the only
transformations that remained, as even rotations had disappeared. But in 2016,
plans were made to reintroduce geometric transformations from the beginning of
secondary school.
That erratic evolution is not without consequence on teachers’ mathematical
culture. Indeed, new teachers face the challenge of having to teach subjects they do

1
The French curriculum is set by the central government and official instructions are published in
the Journal Officiel. Our short summary on the evolution of the teaching of geometric transfor-
mations is based on this material.
8 A. Kuzniak

not really know well and from which they do not master even elementary tech-
niques. Surprising situations occur when, as in CERME in 2011, researchers from
countries where geometric transformations were just re-introduced in elementary
school were wondering whether it is possible to teach them to young students.
French researchers could only report that it was possible, but that transformations
were just removed from their curriculum.

2.3 Taking into Account the Diversity of the Teaching

of Geometry

2.3.1 What Geometry Is Being Taught?

Before dealing with this question, we need to ask ourselves if there is a place for
geometry, as a discipline clearly identified, in all education systems. Indeed, one of
the main issues of the report on geometry done by the Royal British Society (2001)
was to foster the reappearance of the term geometry in the British curriculum. Prior
to this, the study of geometry had been hidden under the heading “shape, space and
measure.” Similar disappearance is apparent in the PISA assessment, in which
geometry topics are covered up by the designation “space and shape”.
These changes of vocabulary are not harmless as they are not only changes in
vocabulary; rather, they reveal different choices about the nature of the geometry
taught in school. The choices imply either a focus on objects close to reality or on
objects already idealized. The decisions on the type of intended geometry relate to
different conceptions of its role in the education of students, and also, more gen-
erally, on the citizen’s position in society. In the French National Assembly, during
the middle of the nineteenth century, a strong controversy about the nature of the
geometry taught in school pitted the supporters of a geometry oriented towards
immediate applications to the world of work against the defenders of a more
abstract geometry oriented to the training of reasoning (Houdement & Kuzniak,
1999). During the second half of the twentieth century, a third more formal and
modernist approach, based on linear algebra, was briefly, but with great force,
added to the previous two (Gispert, 2002). Thus, over the long term and in a single
country,2 the nature of geometry taught fluctuated widely and issues and goals have
changed dramatically depending on decisions often more ideological and political
than scientific. Observations of the choices made nowadays in various countries
reveal irreconcilable approaches that seem to resurrect the debate mentioned above.

2
This conflicting approach on the teaching of geometry is not typically French. In the US, similar
tensions albeit among four different conceptions exist too (Gonzalez & Herbst, 2006) based on
formal, utilitarian, mathematical or intuitive arguments.
2 Thinking About the Teaching of Geometry Through the Lens … 9

2.3.2 Questions of Style

Anybody that has had the opportunity to observe classroom instruction in a country
other than his or her own must have noticed differences in style that can hardly be
accounted to individual differences. The researchers affiliated with the TIMSS
sub-study on teaching practices in six countries noticed such differences in style,
and they used the notion of “characteristic pedagogical flow” to account for
recurrent and typical styles they observed (Cogan & Schmidt, 1999).
To me, this variety of styles appears when reading Herbst’s historical study
(2002) on two-column proofs in the USA. This way of writing proofs is similar to
nothing existing now in France though it is reminiscent of an old fashioned way
used to write solutions of problems in primary school where operations have to be
separated from explanations of reasoning. Another case of cultural shock appears
too when reading Clanché’s and Sarrazy’s (2002) observation of a first-grade
mathematics lesson in a Kanaka primary school (New Caledonia). This time, the
teacher cannot easily assess the degree of understanding of his students for whom
customary respect for the elders forbids their expression of doubts and reservations
in public and thus they never ask some complementary explanation to the teacher.
The analysis of the classroom session allows the authors to claim that the rela-
tionship between mathematics teaching and students’ everyday life should be
analyzed as rupture or obstacle more than as continuity or facilitation.
Let us consider some different styles through an observation made during a
comparative study on the teaching of geometry in Chile and France (Guzman &
Kuzniak, 2006). Various exercises were given to high school pre-service teachers in
Strasbourg, France and in Valparaiso, Chile. As an illustration, we show two stu-
dents’ work using exactly the same solution method but presenting it in radically
different ways. Both are characteristic of what is expected by their teachers.
In Chile, results are given on a coded drawing and the reasoning used is not
explicitly given in writing. By contrast, in France, a very long and detailed text is
written and no assertion, not even the most trivial, is omitted. This point is clearly
apparent in Fig. 2.1 even if Spanish and French texts are not translated.
Observations of Chilean classrooms show that what is written on the blackboard
during a session is often similar to the student’s written production and only oral
justifications are provided, while in France all arguments have to be written
(Guzman & Kuzniak, 2006). Knipping (2008) also shows differences in the use of
the blackboard and in articulation between the written and the oral in France and
Germany. More generally, Knipping (2008) shows that argumentation and proof3
are not equivalent in both countries; rather they give birth to different ways of
developing geometric work in the same grade.
How can we account for these differences in “style” avoiding, if possible, any
hierarchical comparison based on the idea that one approach is fundamentally better

3
In this essay, I mean proof more generally than mathematical or formal proof and different ways
of arguing or validating are considered.
10 A. Kuzniak

Chilean student’s solution French student’s solution

Fig. 2.1 Comparison of two writing solutions in Chile and France

than the other? In the following, I will propose a way to explore these differences
based on the use of geometric paradigms and the theoretical and methodological
model of Geometric Working Spaces (GWS).

2.4 Various Geometries and Geometric Work

2.4.1 Three Elementary Geometries

Houdement and Kuzniak (1999) introduced the notion of geometric paradigms into
the field of didactics of geometry to account for the differences in styles in geometry
education. To bring out geometric paradigms, three perspectives are used: episte-
mological, historical, and didactical. The assemblage of those perspectives led to
the identification of three paradigms usually named Geometry I (or Natural
Geometry), Geometry II (or Natural Axiomatic Geometry), and Geometry III (or
Formal Axiomatic Geometry). These paradigms—and this is an original feature of
the approach—are not organized in a hierarchy, making one more advanced than
another. Rather, their scopes of work are different and the choice of a path for
solving a problem depends on the purpose of the problem and the solver’s
paradigm.
The paradigm called Geometry I is concerned by the world of practice with
technology. In this geometry, valid assertions are generated using arguments based
upon perception, experiment, and deduction. There is high resemblance between
model and reality and any argument is allowed to justify an assertion and to
convince the audience. Indeed, dynamic and experimental proofs are acceptable in
Geometry I. It appears in line with a conception of mathematics as a toolkit to foster
business and economic activities in which geometry provides tools to solve prob-
lems in everyday life.
2 Thinking About the Teaching of Geometry Through the Lens … 11

The paradigm called Geometry II, whose archetype is classic Euclidean geom-
etry, is built on a model that approaches reality without being fused with it. Once
the axioms are set up, proofs have to be developed within the system of axioms to
be valid. The system of axioms may be left incomplete as the axiomatic process is
dynamic and has modeling at its core.
Both geometries, I and II, have close links to the real world, albeit in varying
ways. In particular, they differ with regard to the type of validation, the nature of
figure (unique and specific in Geometry I, general and definition-based in Geometry
II) and by their work guidelines. To these two Geometries, it is necessary to add
Geometry III, which is usually not present in compulsory schooling, but which is
the implicit reference of mathematics teachers who are trained in advanced math-
ematics. In Geometry III, the system of axioms itself is disconnected from reality,
but central. The system is complete and unconcerned with any possible applications
to the real world. The connection with space is broken and this geometry is more
concerned with logical problems (Kuzniak & Rauscher, 2011).

2.4.2 Geometric Working Spaces

The model of GWS4 was introduced in order to describe and understand the
complexity of geometric work in which students and teachers are effectively
engaged during class sessions. The abstract space thus conceived refers to a
structure organized in a way that allows the analysis of the geometric activity of
individuals who are solving geometric problems. In the case of school mathematics,
these individuals are generally not experts but students, some experienced and
others beginners. The model articulates the epistemological and cognitive aspects of
geometric work in two metaphoric planes, the one of epistemological nature, in
close relationship with mathematical content of the studied area, and the other of
cognitive nature, related to the thinking of individuals solving mathematical tasks.
This complex organization is generally summarized using the two diagrams shown
in Figs. 2.1 and 2.2 (for details, see Kuzniak & Richard, 2014; Kuzniak, Tanguay,
& Elia, 2016):
Three components in interaction are characterized for the purpose of describing
the work in its epistemological dimension, organized according to purely mathe-
matical criteria: a set of concrete and tangible objects, the term representamen is
used to summarize this component; a set of artifacts such as drawing instruments or
software; a theoretical system of reference based on definitions, properties and
theorems.
The cognitive plane of the GWS model is centered on the subject, considered as
a cognitive subject. In close relation to the components of the epistemological level,

4
An extension of this model to the whole of mathematical work has been developed under the
name of Mathematical Working Space (MWS).
12 A. Kuzniak

three cognitive components are introduced as follows: visualization related to

deciphering and interpreting signs; construction depending on the used artifacts and
the associated techniques; proving conveyed through validation processes, and
based on a theoretical frame of reference.
The process of bridging the epistemological plane and the cognitive plane is part
of geometric work according our perspective and can be identified through the lens
of GWSs as three geneses related to each specific dimension in the model: semiotic,
instrumental, and discursive geneses. This set of relationships can be described
proceeding from the elements of the first diagram (Fig. 2.2) which, in addition,
shows the interactions between the two planes with three different dimensions or
geneses: semiotic, instrumental, and discursive. The epistemological and cognitive
planes structure the GWS into two levels and help us understand the circulation of
knowledge within mathematical work. How then, proceeding from here, can stu-
dents articulate the epistemological and cognitive levels in order to do the expected
geometric work? In order to understand this complex process of interrelationships,
the three vertical planes of the diagram are useful and can be identified by the
geneses that they implement: [Sem-Ins], [Ins-Dis], and [Sem-Dis] (Fig. 2.3). The

Fig. 2.2 The geometric

working space diagram

Fig. 2.3 The three vertical

planes in the GWS
2 Thinking About the Teaching of Geometry Through the Lens … 13

precise study and definition of the nature and dynamics of these planes during the
solving of mathematical problems remains a central concern for a deeper under-
standing of the GWS model (Kuzniak, Tanguay, & Elia, 2016).
A GWS exists only through its users, current or potential. Its constitution
depends on the way users combine the cognitive and epistemological planes and
their components for solving geometric problems. It also depends on the cognitive
abilities of a particular user, expert or beginner in geometry. The make-up of a
GWS will vary with the education system (the reference GWS), the school cir-
cumstances (the suitable GWS) and the practitioners (personal GWS).
The framework makes it possible to question in a didactic and scientific—non
ideological—way the teaching and learning of geometry.
What is the geometry aimed at by education systems? What is the selected
paradigm? Does this paradigm get selected or does it emerge from practice in
schooling conditions? How do the different paradigms relate to each other?
Moreover, the nature and composition of the suitable GWS is to be questioned:
What artifacts are used? On which theoretical reference is the implemented geo-
metric work really grounded? Which problems are used as exemplars to lead stu-
dents in geometric work?

2.5 Two Examples Showing the Use of the Framework

In the following, I develop two examples showing the possibilities offered by the
framework to deal with the above questions. I refer the interested reader to various
papers using the framework and its extensions, and, specially, the ZDM
Mathematics Education special issue on Mathematical Working Spaces in
schooling (Kuzniak, Tanguay, & Elia, 2016).

2.5.1 An Example of a Coherent GWS Supported

by Geometry I

To show what a suitable GWS guided by Geometry I is, I use the findings from a
comparative study on the teaching of geometry in France and Chile quoted above
(Guzman & Kuzniak, 2006). Education in Chile is divided into elementary school
(Básica) till Grade 8 and secondary school (Media) till Grade 12. From 1998 on, the
teaching of mathematics has abandoned the focus on abstract ideas which was in
place before and turned into a more concrete and empirical approach. As of today,
the reference GWS is guided by Geometry I. To illustrate this and point out some
differences between France and Chile, let us consider the following exercise taken
from a Grade 10 textbook (Mare Nostrum, 2003).
14 A. Kuzniak

Students starting the chapter on similarity have to solve the following problem,
whose solution is given later in the same chapter:
Alfonso is just coming from a journey in the precordillera where he saw a field with a
quadrilateral shape which interested his family. He wants to estimate its area. For that,
during his journey, he measured, successively, the four sides of the field and he found them
to measure approximately: 300 m, 900 m, 610 m, 440 m. Yet, he does not know how to
find the area.
Working with your classmates, could you help Alfonso and determine the area of the field?
(Mare Nostrum, 2003, p. 92)

As four dimensions are not sufficient to ensure the uniqueness of the quadri-
lateral, the exercise is then completed by the following hint:
We can tell you that, when you were working, Alfonso explained the problem to his friend
Rayen and she asked him to take another measure of the field: the length of a diagonal.
Alfonso has come back with the datum: 630 m.
Has it been done right? Could we help him now, though we could not do it before? (ibid.)

The proof suggested in the book begins with a classical decomposition of the
figure in triangles based on the indications given by the authors. But the more
surprising for a French reader is yet to come: The authors ask students to measure
the missing height directly on the drawing. This way of doing geometry is strictly
forbidden at the comparable level of education in France.
How can we compute the area now? Well, we determine the scale of the drawing, we
measure the indicated height and we obtain the area of each triangle (by multiplying each
length of a base by half of the corresponding height). (ibid.)

In this example, geometric work is done on a sheet of paper and with the scaling
procedures, instruments for drawing and measuring, and a formula for calculating
the area of a triangle. In this first GWS, which I call the measuring GWS, splitting a
drawing of the field into two triangles and measuring altitudes makes it possible to
answer the question in a practical way. In that case, geometric work is clearly
supported by Geometry I and goes back and forth between the real world and a
drawing, which is a schematic depiction of the actual field. Measurement on the
drawing affords the missing data. The activity is logically ended by a calculation
with approximation, which relates to the possibility of measuring accepted in
Geometry I but not Geometry II.
A second GWS, the calculation GWS, supported by Geometry II is possible and
exists in France where the so-called Heron’s formula makes it possible to calculate
the area of a triangle knowing the length of its sides without drawing or mea-
surement. The two GWS share a common general strategy: splitting into two tri-
angles. But they do not share the other means of action, the justifications of these
actions, and the resulting geometric work.
In the example, the first two modeling spaces do not necessarily organize
themselves in a hierarchy where the mathematical model would have preeminence.
The GWS supported by Geometry I allows the problem to be satisfactorily solved
with a limited theoretical apparatus. The GWS supported on Geometry II avoids
2 Thinking About the Teaching of Geometry Through the Lens … 15

drawing and measuring and therefore its accuracy is not limited by the measure-
ment on a reduced scale or the imprecisions of the drawing. The procedure in this
GWS allows automation, for example by way of a program on a calculator. The
measuring GWS favors the use of instruments and therefore their associated
geneses, while the calculation GWS fosters the use of symbolic signs (semiotic
genesis). In both spaces, discursive genesis may be called upon to justify the
procedure used but in a different way, which changes the epistemological nature of
proof.5

2.5.2 Intercept Theorem Current Use or Incompleteness

of the Geometric Work

To illustrate the interest of the GWS model and develop the question of the
completeness of geometric work, we will refer to a classroom session (Nechache,
2014) dedicated to the use of the intercept theorem6 (in French, le théorème de
Thalès, or in German Strahlensatz) in France at Grade 9 where the Geometry II
paradigm is favored by the curriculum. In this session, a restricted use of the
mathematical tool, the theorem, leads to a mathematical work that can be often
deemed incomplete. Nechache’s study (2014) helps to clarify some discrepancies
that often arise between the mathematical work produced by the students and the
work expected by the teachers. Our analysis is supported by the GWS model, which
enables highlighting the dynamic of geometric work through the various planes
determined by the model (Fig. 2.3).
In French education, from the 1980s, the use of the intercept theorem has been
gradually restricted to two typical Thales’ configurations: one named “triangle” and
the other “butterfly” (Figs. 2.4 and 2.5).
During the session observed by Nechache (2014), the teacher asks the students
to solve an exercise, taken from the textbook (Brault et al., 2012, p. 311), with nine
multiple choice questions having three alternative answers. Two figures corre-
sponding to Thales’ “butterfly” configuration are associated with the statement of
the problem.
The nine tasks can be characterized as simple, requiring a few abilities: deter-
mine reduction ratios, check equal ratios, and calculate the lengths of triangle sides.
The last four questions relate to the converse and contrapositive form of the
intercept theorem by referring this time to the second figure (Fig. 2.6b) to identify
the correct parallelism properties. In the textbook, the exercise is designed to train
students to identify key figures associated with the intercept theorem and master
routinized techniques. The cognitive activity is essentially based on visual and

5
See Footnote 3.
6
Also known in English as “basic proportionality theorem;” see https://en.wikipedia.org/wiki/
Intercept_theorem.
16 A. Kuzniak

Fig. 2.4 The “triangle” form

of the intercept theorem

Fig. 2.5 The “butterfly”

form of the intercept theorem

semiotic exploitation of data taken from the diagram: no discursive justification is

expected. The mathematical work is fully located in the [Sem-Ins] plane with use of
Thales’ diagram as a technological tool for calculation.
In the classroom session observed by Nechache (2014), the teacher first asks the
students to investigate the questions for four minutes. Then, he only answers two of
the questions he gave and starts with the first question:
In Fig. 2.6a, the triangle AOM is a reduction of the triangle IOE by ratio: 3/9 or
9/6 or 2/3.
The question is simple, because it can be answered in a very elementary way by
using visual recognition using only the semiotic dimension, as the text specifies that
one triangle is a reduction of the other. Different ways to solve it can be used, all of
2 Thinking About the Teaching of Geometry Through the Lens … 17

Fig. 2.6 a, b The diagrams included with the exercise

which involve solely the semiotic dimension. The mathematical work is confined to
the [Sem-Ins] plane by using the butterfly diagram associated with Thales’ theorem
as a semiotic tool. The analysis of the entire session allows us to check that
students’ mathematical work is also confined and closed on the semiotic axis.
The teacher draws the first figure freehand on the blackboard. Before giving the
solution to the first question, he urges students to remember methods related to the
intercept theorem, which had been studied in an earlier lesson when the theorem
was introduced. The solution of the exercise is temporarily postponed in favor of a
work exclusively concerned with the theoretical referential in the suitable GWS
based on Geometry II that the teacher wants to implement. Later, a student reads the
question and gives the correct answer. The teacher agrees and asks him to justify
the answer. This demand of justification is new and is not part of the initial
problem: The student and all classmates remain silent. The teacher reads the
question again and addresses the students:
Teacher: When we tell you that a triangle is a reduction of another one, does
this not remind you of any property? No theorem? Well that’s a pity, we just
saw it 5 minutes ago. So, which theorem has to be applied when we have
such a configuration?
Faced with the remarkable silence of these students who, at this level of
schooling, only know two theorems (the Pythagorean and intercept theorems), and
given that the intercept theorem has just been the subject of an insistent reminder,
the teacher comes back again to the figure drawn on the blackboard by commenting
on it, then he proceeds to checking each of the conditions required to apply the
intercept theorem. He favors the discursive axis in the GWS model by changing the
nature of the task: a justification of the result is requested and needs to be based on a
theoretical tool. The mathematical work has changed and is now in the [Sem-Dis]
plane. The teacher starts by checking the trivial alignment of the points and the fact
that straight lines are transversals (secants in French).
Teacher: Are you sure? Do you have what is needed? How are the points
supposed to be?
Students: Aligned.
18 A. Kuzniak

Teacher: So, the straight lines must be sec…

Students: Secants
Teacher: Which one?
Student: (ME) and (AI).
Teacher: (ME) and (AI) are secants in O. We have the five points which
intervene.
To move forward toward the solution, the teacher resorts to the Topaze effect
that Brousseau (1986) identified when a teacher endeavors to get the expected
answer from his student through purely linguistic cues, independent of the target
mathematical knowledge. In this instance, the mere utterance of the beginning of
the word “secant” with the phoneme “sec” is sufficient to obtain the right answer
from the student.
The teacher then guides the student to check the parallelism of the straight lines
by using the same effect but with less success because students propose straight
lines different from those that are expected by the teacher. These inappropriate
answers show that students no longer perceive the goal of the exercise: They persist
in carrying out a visual work that is not guided by the theoretical referential. But the
teacher remains in his role: He is in charge of developing the theoretical referential
and he finishes by applying the theorem to show equal ratios.
The teacher concludes the session by clarifying briefly what he expects from a
mathematical work.
Teacher: The trick is to be able to explain what we have done.
So the teacher has chosen to adapt the task by changing the nature of the
geometric work: The results should be justified by using the theoretical referential
(the intercept theorem).
In the suitable GWS implemented by the teacher, the mathematical work is
placed in the [Sem-Dis] plane oriented towards the discursive genesis. The expected
validation favors the use of the intercept theorem as a theoretical tool confined in
the discursive dimension of the GWS (Fig. 2.7).

Fig. 2.7 Work done by students versus work expected by the teacher
2 Thinking About the Teaching of Geometry Through the Lens … 19

The observation of this geometry session shows that students’ work is exclu-
sively located in the semiotic dimension favored by the textbook’s suitable GWS
and not expected in the suitable GWS implemented by the teacher. Hence, a
misunderstanding emerges between the work the students do and the work the
teacher expects: The misunderstanding relates to the change of validation in what
counts as proof. Indeed, no discourse of proof is expected in the textbook, but the
teacher does expect proof to be connected to the discourse in the suitable GWS.
Both students and teacher carry out their work diligently, but they do not do the
same geometric work and this work is incomplete because it is confined to only one
or two dimensions instead of all three dimensions of the GWS model.

2.6 Understanding and Developing Geometric Work

Through Its Dynamics

The geometric work perspective that I suggest requires coordination between

cognitive and epistemological approaches, and the entire work is structured by three
complementary dimensions: semiotic, instrumental, and discursive. The research
challenge is to identify and understand the dynamics of geometric work by
observing, in particular, the role of each of the three previous dimensions, and the
interactions among them as suggested by each of the planes used to represent the
model (Figs. 2.2 and 2.3). The successful achievement of this program passes
through a better understanding of each dimension of the GWS model.
Geometry is traditionally viewed as work on geometric configurations that are
both tangible signs and abstract mathematical objects. Parzysz (1988) has clearly
identified this difference under the opposition drawing versus figure, which high-
lights the strong interactions existing between semiotic and discursive dimensions.
In the GWS framework, the semiotic genesis is clearly associated to interpreting
and developing a system of signs (semiotic system) and it could be analyzed using
the contributions of Duval (2006), who developed very powerful tools (in partic-
ular, the notion of registers of semiotic representation) to explore the question. In
his view, a real understanding of mathematical objects requires the student to be
able to play between different registers, which are the sole tangible and visible
representations of the mathematical objects.
Geometry could not exist without drawing tools and study of their different uses
makes it possible to identify two types of geometry, which are well described by the
Geometry I and Geometry II paradigms. From precise but wrong constructions (like
Dürer’s pentagon) to exact but imprecise constructions (like Euclid’s pentagon), it
is possible to see all the epistemic conflicts that distinguish constructions based on
approximation from constructions based on purely deductive arguments. This
fundamental difference continues to nourish misunderstandings and polemics in the
classroom as the “flattened triangle” task shows: Does there exist a triangle with
sides 4, 5, and 9 cm? Some students affirm its existence based on a triangle they
20 A. Kuzniak

have constructed with their compass, and others negate its existence by using the
triangle inequality and calculation.
The tension between precise and exact constructions has been renewed with the
appearance of dynamic geometry software (DGS). As Straesser (2002) suggested,
we need to think more about the nature of the geometry embedded in tools, and
reconsider the traditional opposition between practical and theoretical aspects of
geometry. Software stretches boundaries of graphic precision, and finally, ends by
convincing users of the validity of their results. Proof work does not remain simply
formal, and forms of argumentation are enriched by experiments, which give new
meaning to the classic epistemological distinction between iconic and non-iconic
reasoning. The first closely depends on diagram and its construction and relates to
the [Sem-Ins] plane and the second tends to be based on a discursive dimension
slightly guided by some semiotic aspects [Dis-Sem].
How do the semiotic, instrumental, and discursive geneses relate to each other,
and specifically how does the use of new instruments interact with semiotic and
discursive geneses in transforming discovery and validation methods? And how can
students’ geometric work be structured in a rich and powerful way? This is one of
the issues that the GWS model seeks to describe through the notion of complete
geometric work Kuzniak, Nechache, and Drouhard (2016a) which supposes a
genuine relationship between the epistemological and cognitive planes and artic-
ulation of a rich diversity between the different geneses and vertical planes of the
GWS model. The aim is not only to observe and describe existing activities but also
to develop some tasks and implement them in classroom for integrating the three
dimensions of the model into a complete understanding of geometric work
according to the perspective expected by teachers and that geometric paradigms
help to precise.

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geometry figures. Educational Studies in Mathematics, 19(1), 79–92.
Royal Society/Joint Mathematical Council. (2001). Teaching and learning geometry. London:
Royal Society/Joint Mathematical Council.
Straesser, R. (2002). Cabri-géomètre: Does dynamic geometry software (DGS) change geometry
and its teaching and learning? International Journal of Computers for Mathematical Learning,
6(3), 319–333.
Sinclair, N., Bartolini Bussi, M. G., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., et al.
(2016). Recent research on geometry education: An ICME-13 survey team report. ZDM
Mathematics Education, 48(5), 691–719.
Chapter 3
Epistemological Features
of a Constructional Approach
to Regular 4-Polytopes

Stephan Berendonk and Marc Sauerwein

Abstract The limitations of empirical methods in 4-dimensional geometry

demand, but also provide more scope for, alternative ways of understanding, such
as analogy. The introduction of students to 4-dimensional objects should thus be
considered as an opportunity to enhance students’ belief system about mathematics.
We will describe and reflect on the characteristics of a constructive approach to
regular 4-polytopes and share our experiences with teaching this approach.

Keywords 4-Polytopes Analogy Beliefs Constructional Empiricism





Four-dimensional Geometry Hypercube Induction Mental object




Platonic solids Reasoning by analogy Subject matter didactics
Workshop

3.1 A Didactical Challenge: The Pure Empiricist

Consider the following task: Show that the graph of a quadratic function is a curve
whose points are at equal distance from a fixed point and a fixed line.
“How to accomplish this task? I know. I open my dynamic geometry software
program and let it plot the graph of some quadratic function. Fortunately, there is
also a button that creates, by specifying a point and a line, the curve whose points
are at equal distance from them. So, I have a second curve on my screen; and, by
dragging the specified point and line, I can manage to put that new curve right on
top of the old one. Thus, indeed, both curves are exactly the same. QED.”
If someone actually solved the above task in the described way, he or she would
be convinced that both constructions yield the same curve. Thus, it would be

S. Berendonk
University of Duisburg-Essen, Essen, Germany
e-mail: stephan.berendonk@uni-due.de
M. Sauerwein (&)
University of Bonn, Bonn, Germany
e-mail: sauerwein@math.uni-bonn.de

© Springer International Publishing AG, part of Springer Nature 2018 23

P. Herbst et al. (eds.), International Perspectives on the Teaching and Learning
of Geometry in Secondary Schools, ICME-13 Monographs,
https://doi.org/10.1007/978-3-319-77476-3_3
Exploring the Variety of Random
Documents with Different Content
 BIOGEAPHICAL SKETCHES. 1091 esiiiration of said
enlistment he returned home and engaged in the carpenter business
at Picture Kocks, Lycoming county, Pennsylvania, for a short time. In
August, 1802, he again enlisted, in Company H, One Hundred and
Thirty-first Pennsylvania Volunteers, and held the rank of second
sergeant imtil mustered out at the end of nine months. He at once
entered the United States Militia Railroad department as a carpenter
and bridge builder, with headquarters at Alexandria, Virginia, and
served in that department until after the close of the war. He was
married, June 21, 1S64, to Sarah E. Wallize. of Xorthumberland
county, Pennsylvania, and to this union have been born: Flora I.,
wife of O. H. Hilliard; Dora E., wife of John L. Fisher; Ambrose P.;
Sherman S. ; Lewis W., and Loreda M. In ISfiS he came to
Watsonto\vn and accepted his present position. Mr. and Mrs. Kistner
are members of the Lutheran church, in which he has served as an
active member in the church coimcil. He is a member of Bryson Post,
No. 22-5. G. A. R., and was its commander for 1890. He is a
Republican in politics, and has served a number of years in the town
council, and also as school director. Joseph Ely, machinist, was born,
January 17, 1830. in Union coimty, Pennsylvania, son of John and
Lydia (Rhoades) Ely. Our subject was educated in the common
schools and at the age of eighteen years began to learn the trade of
machinist. He was engineer in a saw mill for Seth T. McCormick for
some time, after which ho assisted his father in rtinning a canal
boat. He helped to construct the first saw mill in Watsontown, in
which he was employed until he enlisted in Captain David Ely's
company, known as Company E, One Hundred and Thirty-first
Pennsylvania Volimteers. August 8, 1802. After serving out his term
of enlistment he again enlisted, September 0, 1801, in Company D,
Two Hundred and Second Pennsylvania Volunteers, and remained in
active service until the close of the war. After returning from the war,
he, in i)artnership with the firm of FoUmer A: Cook, ojierated a
planing mill for one year, when the mill was destroyed by fire, after
which time he was connected with ditterent lumber companies of
"Watsontown until 1870, when he was employed by the Watsontown
Planing Mill Company, in which institution he is still working. In 1872,
when the first fire engine was purchased by the citizens of
Watsontown, he was chosen engineer, and in 1875 was appointed
chief engineer, and has served as such ever since. He was married,
December HO, 18.38, to Harriet Slenker, who died, January 17,
ISOS. To this union were born three children, one of whom is living,
Ida, wife of James Findley. of Iowa. He was again married, June 17,
1809, to a Mrs. Hogey, whose first husband died in the army. By this
union two children have been born: Irman I. and Minnie, both
deceased. Mrs. Ely had two children by her first marriage: Charles
and Mary C. Mr. and Mrs. Bly are consistent members of the
Reformed church of Watsontown.
 1092 HISTORY OF NORTHUMBERLAND COUNTY. W. A.
Durham was bom in Northumberland county, Pennsylvania, January
7, 1852, son of Joseph G. Durham. He was educated in the common
schools and at Dewart Academy. He taught school three years, and
then engaged in the mercantile business in Watsontown. Soon after
he opened a general store at Coburn, Centre county, Pennsylvania,
which still continues in the name of Durham Brothers & Company.
From 18S0 to 1SS4 he dealt in farm produce, agricultural
implements, etc., in Watsontown. Since 18S7 he has been engaged
in the lumber business and in manufacturing doors, sash, and blinds.
May 31, 1876, he married Mary A. Giffen, of Marion, Iowa, and to
this union have been born four children: Joseph E., deceased; Glenn
G. ; W. Leigh, and James G. Mr. and Mrs. Durham are members of
the Presbyterian church of Watsontown, of which church he is a
trustee. He is connected with the Masonic order, is secretary of the
to^vn council, and in politics is a stanch Republican. Alfred Hockley,
manufacturer and dealer in carbonated drinks, was born in
Montgomery county, Pennsylvania, son of Clement L. and Elizabeth
(Keyser) Hockley, natives of that county. The father was a farmer by
occupation, and was one of the first men to vote for the public
school system in Montgomery county. He reared a family of nine
children: Alfred; Jacob, agent for the Standard Pipe Line Company of
Ohio; William, a school teacher, of Lycoming county; Irwin R., who
has been principal of the high school at Emporium, Cameron county,
ten years, but is now in the general merchandise business; Franklin
C, deceased; Theodore; Amelia; Lucy, and Maggie. The subject of
this sketch was educated in the public schools of Montour county,
and worked as a carpenter ten years. He then opened a grocery and
provision store in Mahanoy City, and remained there eight years, and
three years was at the same business at Eldred, McKean county,
Pennsylvania. In 1880 he came to Watsontown, and has since been
engaged in the manufacture of carbonated drinks. He was married,
December 1, 1870, to Hattie Fox, of McEwensville, and they are the
parents of three children: Roscoe I., deceased; Chester F., and
George A. Mr. and Mrs. Hockley are members of the Lutheran
church; he is a member of the town council, and on the 1st of June,
1S*J0, became chief burgess of Watsontown, being appointed by
the town council vice E. Sherman Follmer, resigned.
 BIOGRAPHICAL SKETCHES. 1093 CHAPTER XLVII.
BIOGEAPHICAL SKETCHES. XORTIIUMBEKLAXD BOROUGH AND
PoIXT ToWXSIIIP. Thomas Taggart was the lirst of the Taggart family
to come to Northumberland county. He was Njrn in Ireland, May 10,
172S, and prior to the year 1750 immigrated with his brother,
Robert, to Bhiladelphia, where the latter became a merchant. It was
about 1775 when Thomas made his advent into the toWn of
Northumberland, settling near the present site of Morgan's shoe
store. He marrit-^1 Man.- Tanderbilt, a native of Philadelphia, who
died in Northumberland in IS' '5. The children of this union were:
Elizabeth, who was born. January 15. 1751:!. married William
Bonham, and died al)out 17S0 (her son. Thomas, was for many
years a tanner at Northumberland, but finally removed to Wabash
county, Illinois, where he died); Christiana, who was born. May 17.
1755. married a Mr. Sample, and settled in Allegheny county, this
State: Robert, born, February IS. 1757: John. who was born, June
;>(). 175'.). and died, July 21, 1759; Catharine, who was born,
September G, 1760, married Captain John Painter, and died in 18-tO;
Thomas, who was born, October 27, 17'J2. and died, January 16,
17S0; Mary, who was born, January 19, 1765, and married a Mr.
Patterson, a noted frontiersman of Pennsylvania; John, who was
born, July 11, 1767, and died, Febntary 8, 177-5; David, who was
born. February 21, 1769, and died, May 17, 1S12; A\'illiam. who was
born, Octoljer 8. 1771, and died, January 24, 1778: William (2d),
who was born. Augu-t 6. 1778, kept store at an early day in
Northumberland, and in the latter part of his life resided on a farm in
Chillisquaque township, where he died at the age of eighty years,
and James, born. January 1, 1780. David Taggart, pre\-iously
mentioned, was educated in Northumberland and was a prominent
Democrat. He married Marv' McCalla and to them were born the
following children: John, born. April 12, 1796, in Northumberland;
James, who died in Xorthimiberland about 1855, was a merchant,
and was engaged for some time in running ])ackets on the canal,
and was collector of tolls on the canal at Himtingdon for many years
(his son, Grantham I., is a coal dealer at Savannah, Georgia, and
another son, John, deceased, was a physician and died at Salt Lake
City; his two daughters were: Mary, who married Marks B. Priestley,
and Gertrude, who married Solomon
 1094 HISTORY OF NORTHUMBERLAND COUNTY. Kregar);
Sarah, who married Samuel C. McCormick, and Mary A., who
married Alexander Colt. John Taggart was educated in
Northumberland county and began his business career as a brewer
at the town of Northiimberland, his brewery being located near the
present steamboat landing. When the canal was constn.icted his
brewery was removed to give place to it. so he quit the business. He
was appointed canal commissioner by Governor Joseph Eitner, held
the position about one year, resigned, and was succeeded by
Thaddeus Stejihens. He was president of the Northumberland Bank
for a number of years. He married Hannah Huston, a native of
Philadelphia. Both were members of the Unitarian church, and at the
time of his death Mr. Taggart was a Eepublican in politics. He was a
highly respected, enterprising citizen. David Taggart, son of John,
received a good education, read law with Ebenezer Greenough, wfts
president of the Pennsylvania State Agricultural Society for some
years, was State Senator before the war, serving the last year of his
term as Speaker of that body; he was chairman of the Whig State
central committee in 1852, subsequently a Republican, and was in
the paymaster's department during the war. He married Anna P.
Cowden, and his children were: James; Helen, and Hannah. Matthew
H. Taggart, son of John, was brought up in Northumberland, iKirn,
February IS, 1824, and was educated in the public schools and
Lewisburg Academy, closing his literary studies at the latter in 1842.
He began at once to clerk for Walls & Green, general merchants,
then at Lewisburg, where he remained for three years, after which
he commenced the study of law with his older brother, David.
Subsequently he taught school for two winters and then took charge
of a store in Union county, remaining there from 1840 to 1858.
Closing this connection, he opened up a store on his o^vn account
at Beaver furnace. Union county, this State, where he did a
successful business for six years, removing thence to
Northumberland, and soon became an employee of the State
treasurer's ofKce at Harrisburg, continuing in that important position
for about ten years, having his home a part of the time in Lancaster
county. In 1884 he removed to Philadelphia, where he resided until
1890, when he returned to his native town of Northumberland to
give personal attention to the business of the Taggart Nail Mills, in
which he has been interested for fifteen years. His first wife was
Eebecca McCurley, who died, leaving six children: Harrj' C. and John
K., both interested in the nail mills; Anna, widow of J. F. Frueauff;
Clara, and James, book-keeper at Godcharles's nail mills at Milton.
His second wife was Eliza McCurley, by whom he has the following
living children: Matthew H., Jr., and Mary. His third wife was Ella G.
Royer before their marriage. Captain James Tilggart, third son of
John Taggart. was born in the borough of Northiunberland, February
4, 1827, and received his education almost entirely at the public
schools of his native town. At the age of six 
 BIOGRAPHICAL SKETCHES. 1095 teou years he was
employed by the hite Ejjhraim Shannon in his store at
NorthumberLind, where lie remained four years, properly mastering
the duties assigned to him and to a large extent engrafting the spirit
of his employer, whose genial humor, facetious witticisms, and well
spun yarns are thoroughly remembered by the older citizens who
congregated around this store, where fun always took precedence.
In 1848 he went to western New York in the capacity of paymaster
for James Moore, who had a large contract in the vicinity of
Hornellsville, and after completing this he returned to
Northumberland, purchased the store of the late William H. Waples,
and entered into business for himself. He forgot none of the training
by his old employer, and mixing fun with business he did a fairly
profitable trade until 18ni, when he entered the army. In December,
1850, he married Sarah, daiTghter of John H. Cowden, by whom he
had four children; two died in infancy and two survive to the present
time: Sarah C, who resides in Northumberland, and David, a
physician at Fruckville, Schuylkill county, this State. Immediately
upon the news of the tiring ou Fort Sumter, a roll was started,
naturally at his store, because it was the i)riucipal rendezvous for the
yoimg and active spirits of the community. A comjVany was formed
and he was elected cajttain. But. owing to the spontaneous response
to the call of President Lincoln and the inability of the government to
furnish guns for all, this eager company was not directly accepted;
but through the foresight of Governor Curt in in effectively
recommending the formation of reserve corps, tliey were received
a> Conipauy B. Fiftli Pennsylvania Reserve, and placed under
command of that brave and able officer. Colonel Seneca G.
Simmons. This comjiany has a history and the subject of this sketch
figures largely in that history. His comrades will attest his worth as a
man and soldier; and the laying down of his life, as he did in the
field at the battle of Charles City Cross Roads, proved his valor and
patriotism. John K. Taggart, another son of John, was secretary and
clerk to his brother, David, in the paymaster's department in the
army, and died about 1808 in St. Louis. Hannah, daughter of John
Taggart. married Dr. Joseph Priestly and has four children: Hannah,
who married Rev. H. D. Catlin; Fannie D., wife of William Forsyth, Jr.;
Anna, and Jennie. Frank A. Taggart, son of John Taggart, is
employed by the Pennsylvania Railroad Company in PhiladeliDhia,
and his children are: John; William; Joseph, and Anna, Mrs. Christy.
William Forsyth, deceased, was born of Scotch parentage in
Hamburg, Berks coimty, Pennsylvania, October 8, 1788. His father,
Andrew Forsyth, was with Washington in Philadelphia and Valley
Forge. By reason of depreciation of Continental money he was
reduced in circimistances, and with his family moved to the mouth of
Fishing creek, where he taught school and also at Danville. In 1804
William was apprenticed to John Cow 
 1096 HISTORY OF NORTHUMBERLAND COUNTY. den, of
Northumberland, to learn the art and -mystery of store-keeping.
After he was free he managed a keel-boat, and was one of the first
to navigate the Susquehanna river from Northumberland to Owego,
New York, carrying iron and stoves to the latter, and Ijringing salt
and plaster to the former. In ilarch. 1817, he was married to Betsey
Gaskins, daughter of Thomas Gaskins of Point township, and kept a
tavern and stage office on Water street, Northumberland. At the
commencement of the construction of the public works, he took
contracts and built several miles of canal, and also of the Sus, IS 14.
son of Alexander and Mary Elliott, born, respectively. May 29, 1777,
and October 31, 1783. The father was of Scotch-Irish extraction, and
after learning the hatter's trade at Eeading, Pennsylvania, removed
from tha,t city to Northumberland county and followed his trade at
the borough of Northumberland, where he died, February .">, l,s;',ri.
To Alexandi-r and Mary Elliott were born the following children:
Mary, born July 4, 1804: Thomas, bom September 18, 1800;
Margaret, born August '21, ISIO; Sarah; William: Ann, and Samuel.
Our subject received a common school education with means
obtained l)y labor at such jobs as he could get to do. The first work
he did for himself, was digging in a ditch from the river to what is
known as the gut in Sunbury — the old Sunbury canal, which was
never comjileted. He received seventy-five cents at this employment
and was subsequently hired by John Cowden for one summer at two
dollars fifty cents per month. With the money thus obtained he
attended school, doing chores for William Forsyth mornings and
evenings for his board and lodging. After closing this term of school
he took employment at the tavern of C. Buttles, in Northumberland,
where he remained one summer, and then drove a team on a trip to
Philadelphia with William A. Lloyd. After this he jjurchased a team
and followed farming and teaming for some time. He was married in
January, 1837, to Christiana Knox, and settled in Northumberland.
Two years later he removed to a farm of one hundred acres on what
is kno^vn as Blue Hill and which still belongs to the family. In 1841
he began boating on the canal, transporting lumber, coal, etc., which
he continued successfully eight years. He then purchased an interest
in the grocery business of Henry Wenck; two years later he became
sole proprietor of the store, which he conducted until the
Philadelphia and Erie railroad was opened, when he withdrew and
took the position of first station agent for that company at
Northumberland; after this began a series of railroad contracting, as
follows: Bennett's branch; Oil Creek road, Venango county;
Philadelphia and Erie, at Will
 BIOGRAPHICAL SKETCHES. 1097 iamsport, eight miles of
double track, and eight miles of the same road fnjm St. Mary's to
Rathburn. He was appointed by the State canal coumiissioners as
superintendent of the canal from the mouth of the Juniata river to
Northumberland, to which was subsequently added the line to
Mihon. In l'^^n he was elected to the legislature from this county;
he has also served as school director twentyfive years, and belongs
to the I. O. 0. F. His wife died. November 7, 1875, and was the
mother of four children: Anna: Emily I., wife of Charles Davis; Mary
J., wife of A. H. Hathaway, and John. The last named was educated
at Freeburg College, graduated from the Bellevue Hospital of New
York City, and is practicing medicine in Northiunberland. Mr. Elliott is
the owner of one hundred seventy-six acres of land in Fnion county,
three hundred seventy-eight acres in Chillisquaque township, one
hundred thirty-eight acres in Point township, and other valuable
property, all of which is the acquisition of his own labors. Dr. Robert
McCay. dfCcased. was burn in Nurthumberlaiid, May 17, LS20. His
father. William MeCay, was born in ]77t; and was, it is "said, the first
white child born in the town of Northumberland. His grandfather,
Neal McCay, was of Scotch nationality; he immigrated to America,
settled in New Jersey, and about 1774 removed to Northiimberland,
where he cleared a tract of land. He died, March 2, 1S14. William
McCay became a land speculator and a practical surveyor. He served
in the war of 1812. He was twice married, and lived and died in
Northumberland. Dr. Robert McCay, the subject of this sketch, was
educated at Gettysbiirg, read medicine with Doctor McClellan, of
Philadelphia, and graduated from the University of New York in the
winter of 1844— 4.1, after which he practiced his profession in
Danville one year, and then located at Northumberland, where he
had an extensive practive until his death. He was a tine Spanish
scholar, aiL
 1098 HISTORY OF NORTHUMBERLAND COUNTY. William
Stoner, graduated from the Woman's Medical College of Philadeljjhia
in 1887, and has since practiced in Northumberland and Sunbury.
She was married in LST."). Mrs. McCay, the widow of our subject,
has been engaged in the drug business for several years. This drug
store was established by Miss Mary McCay, sister of Dr. Robert
McCay, in 1848, and has since remained in the family. Je.sse C.
Hoetox, deceased, was bom near Forty Fort, Luzerne county,
Pennsylvania, May 1, 1797, son of John and Mary (de la Montague)
Horton. The latter was of Scotch and French descent; her mother,
whose maiden name was Sarah Miller, was a Scotch Puritan, and her
father was the son of a French physician. John Horton was of English
descent. During the Revolution he served as a lieutenant in the New
Jersey militia. After the conclusion of peace he sold his property in
that State for Continental money and removed to the vicinity of Forty
Fort in the Wyoming valley, Luzerne county, Pennsylvania. There he
died in 1810, leaving a widow and nine children. Jesse C. was then
in his thirteenth year. Four years later he served in the New Jersey
militia under Colonel Seward and Captain Swazey in the war of 1812.
In 1816 Miller, Lewis, and Jesse C. Horton inaugurated a new era in
stage coach traveling in northern Pennsylvania by establishing a line
of four-horse coaches from Baltimore to Owego, New York, by the
way of Harrisburg. Sunbury, Wilkesbarre, and Montrose, and also a
line from Philadelphia to Wilkesbarre by the way of Easton, and from
New York City to Montrose, Pennsylvania, by the way of Morristown
and Newark, New Jersey, and Milford, Pennsylvania, with postoffices
established at Plymouth, Kingston, Pottstown, and Tunkliaiinock. In
1820 he lived at Berwick, Columbia county, Pennsylvania, engaged in
staging and carrying the United States mail; he removed from there
to Owego, New York, in 1881 to Northumberland, and soon after to
a farm in Point township. In 1841 he represented Northumberland
county in the legislature, and in 1842 he was elected to the Senate.
He was a stockholder and director in the Bank of Northumberland.
FebruaiT 6, 1820, he married Harriet Ford, daughter of Dr. Samuel
Headley, of Benvick, by whom he had two children: Annie Maria,
Mrs. Allen M. Gougewer, of Washington, and Mrs. Harriet Westler, of
Berwick. His wife died, June 17, 1823. He was again married,
December 12, 1825, to Mrs. Martha Cooke, daughter of James
Lemon, of Northumberland coimty. Five children were born to this
union, two of whom are living: Mary C, and Amelia H., wife of
Anthony Simpson, of Michigan. The second wife died, July 25, 1S80.
Mr. Horton was a Democrat in politics. He was active in getting the
public school system established and an active worker and supporter
of any enterprise conducive to the welfare of the general public. He
was well kno\vn throughout Pennsylvania, and had an extensive and
intimate acquaintance with the prominent men of the State.
The text on this page is estimated to be only 0.00%
accurate

^£r^,^^^
 BIOGRAPHICAL SKETCHES. . 1101 C. W. GuTELU-s, editor
niul publisher of the Public Press, was Ijorn in Mifflinburg, Union
county, Pennsylvania, November 4, 1S:57, son of Israel and Sarah
Gutelius. The father was bom in ]snrchased in 1884 the general
store of William T. Forsyth at Northumberland and continued the
business until 188(1 In LSS-") he was appointed postmaster at
Northumberland and served in that office until IN'.M). He was
married in ISdri to Alice Gulick, by whom he has four children:
Charles G. ; \\ii\Wv L. : Jennie C, and J. " Howard. He served as
chief burgess of Northumberland for three successive years, and is a
Democrat in politics and a member of the Lutheran church. S. M. G.
Wenck, pharmacist, was born in the borough of Northumberland,
September 24, 1830, son of George and Elizabeth (Pardoe) Wenck,
of German and Quaker ancestry, respectively. He received an
ordinary education, and in 18o5 established his 2)resent business at
McEwensville, this county, where he remained until 1880; he then
removed his stock to Northumberland, where he has built up a large
trade. He was married in 1855 to Christiana Kauffman, by whom he
has three children: William G. ; Maggie, wife of Morton McFarland,
and Edgar S. Mr. Wenck is a member of the Lutheran church. James
Dieffenbacher was born in a part of Northumberland (now Montour)
county, Pennsylvania, May 31, ISOlJ, son of Conrad Diett'enbacher, a
native of Montgomery county, this State, and a grand.son of Conrad
Dief62
 1102 HISTORY OF XORTHUMBEHLAND COUNTY. fenbaclier,
who was horn in Germany and settled and died on the farm where
our subject now resides. Our subject learned the carpenter trade,
which he followed forty years. He settled in Northumberland in 1831,
and was married in 1S3S to Nancy Goston, who died in 1842, the
mother of one child, James G. He was again married to Maria
Hunsicker. He was a justice of the peace twenty-two years and a
school director twenty years. Jacob M.A.TTHi.is, merchant, was born
iu York county, Pennsylvania, September 20, 1828, sou of John and
Elizabeth (Growl) Matthias, natives of York and Lancaster counties,
respectively. They moved to Daujihin county, Pennsylvania, where
both died. They were members of the Lutheran church. Jacob
Matthias was reared in York county, and attended the local schools;
he has been engaged in various occupations, such as boating,
driving stage, and railroading. In 1860 he removed to
Northumberland county and established his present business. In
1850 he married Susan Van Dyke, daughter of Israel Gross, of
Snyder county. Their children, by adoption, are: Mary Guistwelt, wife
of Harry Frey, of Cumberland county; Susan Gross, wife of Frank
Shilmer, of Northumberland, and Bertha Newman. Mr. Matthias and
wife are members of the Lutheran church; in politics he is a
Republican, and has served as overseer of the poor sixteen years.
Cyrus Broi'se. merchant, was born in that part of Union county which
is now Snyder county, Pennsylvania, October 15, 1840, son of Peter
and Amelia (Moore) Brouse, both natives of that county, and farmers
by occupation. They were members of the Lutheran church. The
mother died in 1887, and the father in 1889. They reared five
children, four of whom an^ living:Cyrus; Margaret, Mrs. E. I. Snyder,
of Lewisburg, Pennsylvania; Mary E., Mrs. Abel Johnson, of Union
county, and Franklin, of Sunbury. The subject of this sketch was
educated at the township schools and at Freeburg Academy. In
September, 1802, he came to Northumberland, and first engaged as
clerk for William T. Forsyth, with whom he remained until June,
1803; he then engaged with M. H. Taggart, and was subsequently in
the employ of Reuben Johnson, Colt & Todd, and others. April 1,
1888, he established his present business. In 18*54 he married
Esther J., daughter of John Johnson, of Point township, by whom he
has five children: Reuben; Franklin C, who married Sarah Dull, of
Lewisburg; Thomas; Williard, and Earl M. Politically Mr. Brouse is a
Republican, and has served as councilman and burgess of
Northumberland; he is a member of the K. of P. of Northumberland.
He and family are members of the Methodist Episcopal church, in
which he holds the office of recording steward. W1LLI.A.M B. Stoner,
physician and surgeon, was born at Westminster, Carroll county.
Maryland, October 2('), 1845. He received his education at a
collegiate institute at that place, now known as the Western
Maryland College, also received four years' instruction under
Professor Beggs, now of Belfast, Ireland, and graduated from the
Edinburgh University. After com 
 BIOGRAPHICAL SKETCHES. 1103 pleting his education he
read medicine with Dr. William Taylor, of Gettysburg, Pennsylvania,
was graduated from the Philadelphia University' of Medicine and
Surp^ery in ISfiC), and began practicing in Hummelstowu, Dauphin
county, where he remained until ISTI. From there he removed to
Georgetown. Northumberland county, and in IST:^ to
Northumberland, where he has since enjoyed an extensive practice.
His father, George W. Stoner, was a native of Dauphin county. He
married Mary A. Sullivan, and settled in Carroll county. Maryland,
where he engaged in farming and still resides. They reared ten
children, seven of whom are living: William B. : Georo-e W., of
Baltimore. Maryland; Annie, wife of John C. Eckenrode, of
Westminster, Maryland: Sarah S.. wife of Martin Babylon, of
Westminster; Leah S., wife of William C. Robinson, of Littlestown,
Pennsylvania; James M.. of Baltimore, and Maggie. Doctor Stoner is
a member of Eureka Lodge, F.
 1104 HISTORY OF NORTHUMBERLAND COUNTY.
(Farnsworth) Reed. His father wa.s born in IT'JO in Shamokiu
township, where his father had settled, and where he lived and died.
After his marriage he located in Rush township, but in 1833 moved
back to Shamokin townshij). He was a Whig in politics, and a deacon
in the Baptist church for many years. He died in Shamokin township
in LSoU, and his wife in 1857. They reared seven children, all
deceased except Farnsworth and Sarah, Mrs. William Depew, of
Riverside. The subject of our sketch attended the schools of Rush
and Shamokin townships. In 1843 he married Rosanna, daughter of
David Miller, of Shamokin township, and has since resided in
Shamokin, Augusta, and Point townships, settling on his present
farm in the spring of 180G. His family consists of nine children: S.
O.. of Suiibury; Maria Elizabeth, Mrs. Charles P. Seasholtz, of
Northiuuberland; Clarissa A., Mrs. George W. Vandevender. of Snyder
county: Jacob A., a merchant of Winfield, Pennsylvania; Sarah L.,
Mrs. Thomas J. Tandelbing, of Packer's island; Laura D., Mrs. Charles
M. Park, of Kingston. Peimsylvania; Harriet I., Mrs. James B. Leslier,
of Nebraska: Elmer E.. of Point township, and George M., of Iowa.
Mr. Reed is an active memlier of the Republican party, and has
served in various township otKcps. He is a deacon in the Baptist
churcli of Xorthiimberland. Geoiuie M. Ditzler, farmer, was born in
Tnrbut township. Ncjnhumberland county, Penusylvauia. January -
">. ISl'.t, son of Jacob and Sarah l Overpeck) Ditzler. He was
educated in the pul)lie schools and learned the trade of tinsmith,
which occupation he followed for ten years and has since been
farming. In 1S74 he was married to Isadore M., daughter of Stephen
Bittenbender. one of the early settlers of Shamokin, by whom he has
one child. William. He is a member of the Masonic lodge of
Northumberland, the chapter of Danville, and also of the Danville
commandery. He is a Republican, and a member of the Lutheran
church. He settled upju his present farm in 187.") and is bringing it
to a high state of cultivation. H.\KEisoN C. Kase, farmer, was Ijorn in
Rush township, Northumberland county, Pennsylvania, April 2U,
1N44. His father, Charles Kase. was also born in Rush township. His
grandfather, John Kase, was a native of England, who came to
America as a young man, married Eleanor Dewitt, a native of New
Jersey, and was among the early settlers of Rush to^^Tiship. He
and his wife were members of the Presbyterian church for many
years. They retired from farming and moved to Ely.sburg, where
both died. They reared eight children, four of whom are living:
Charles, of Riverside: Catharine, Mrs. George West, of Danville;
Clinton, a farmer of Montour county, and James, of Danville. Charles
Kase, the father of our subject, was engaged in farming in Rush
township until he retired and moved to Riverside, where he now
resides. He is a stockholder in the Danville Bridge Company. Mt.
Carmel Bank, and Danville nail factory. His wife died, Augnst '25,
1880. Thev reared four children; Oscar S., of Riverside; Harrison C. ;
 BIOGRAPHICAL SKETCHES. 1105 Hannah Ellen, deceased,
and John Wilson. The subject of this sketch was reared in Eush
township and .^lucated in the township schools and Dan\ille
Academy. In 1SG3 he went as a substitute for his father in Company
H, One Hundred and Seventy-second Penn.sylvania Volunteers, and
served nine months, when he re-enlisted in Company A, Sixteenth
Pennsylvania Cavalry, and served one year; since then he has been
engaged in farming. February 2, 1870, he married Maranda Ellen,
daughter of Robert and Bethiah (Banghart) Davidison, natives of
New Jersey and settlers of Montour county, bv, whom he has three
children: Cora May; Eleanor Gertrude, and Bertha Ellen M. Politically
Mr. Kase is a Eepublican, and has served in the various township
otKces; he is a member of Goodrich Post. G. A. R.. and of the
Presbyterian church of Danville. Robert Cukky McWilliams, farmer,
was born in Mooresburg. Montour county, Pennsylvania, February
20, 1845, son of John and Margaret (Caldwell) McWilliams, of the
same county, and of Scotch-Irish extraction. He received his
education at the juiblic schools, and at academies at Milhille,
Pennsylvania, and Newark, New Jersey. His business career has been
principally that of a farmer, and he came to this county in 1872. In
October, ]87(), he married Louisa V. Reighard, daughter of Dr. Jacob
Reighard. of Juniata county, this State. To this union have been born
six children: Margaret Caldwell; Lucinda AVagner: Amanda Belle;
Jenette June; Robert Curry, and Mary E. ]Mr. ;^Ic^\'illiams has
always been an active worker in the Democratic party; in the winter
of 1883-84 he represented Northumberland county in the State
legislature, and was a member of the committees on military,
election, federal relations, and banking. He has also tilled various
township otlices. He is a member of Eureka Lodge, No. 404, F. &. A.
M., a charter member of Warrior Run Cha]iter, No. 240, and a
member of the S. P. K. uf Northumberland. He is a deacon and elder
in the Presbyterian church of Northiimberlaiid. Henry Watts, farmer,
was born at his present homestead, October 11, 1817. His father,
John Watts, was a native of England, who immigratetl to America in
1801 and settled ui^on the farm on which our subject now resides
in 1 M t2. He married in this country, Sarah Hales, also a native of
England. He was a Whig in politics, and his wife was a member of
the Bai^tist chitrch. He died in ISHO. and his widow in 1854. They
reared three children: Henry: John, and William, of Des Moines,
Iowa. The subject of this sketch was reared and educated in Point
township, and has always lived upon his present farm. In 1847 he
married Lydia Ann, daughter of Peter and Margaret (Giest) Dentler,
of Point township. They have no children. Mr. Watts is an active
Republican, and has served on the election board twenty-five years,
as asses.sor of the township eight years, and as school director a
number of terms. Mr. and Mrs. Watts are members of the Baptist
church of Northumberland.
 lion HISTORY OF NORTHUMBERLAND COUNTY. H. W. Burg,
physicican and surgeon, was born in Lower Windsor, York county,
Pennsylvania, May 10. iSo'i. He received his literary education at
New Berlin, Union county, the W'illiamsport Seminary, and at a
commercial college. He read medicine with Dr. S. W. Burg, of New
Berlin, attended lectures at the Jefferson Medical College, from
which he graduated in 1877, and has since practiced his profession
in Northumberland. In 1875 he married Cora H., daughter of A. C.
Simpson, attorney at law, of Selinsgrove, Pennsylvania, by whom he
has three living children: Edwin; Stoddard, and Dorothy. A daughter,
Mary Blanch, was born, December 23, 1882, and died on the 13th of
October, 1SS-"). The Doctor is a Democrat in politics, has served as
burgess and assessor of Northumberland, and is a member of the
Sunbxtry Medical As.sociation. His wife is a member of the Episcopal
church. His father, Philip William Bur^. was a native of Amsterdam,
Holland, came to America when a young man. and settled in York
county. He married Mary A. Eckert, a native of Nurthmuberland, and
lived in Y'ork county iintil 1855, when he died, and his wife and two
sons removed to Northumberland in the same year. They had two
children: H. W., and Somers, a machinist of Danville. CHAPTP]R
XLVIII. BIOGEAPHICAL SKETCHES. TLU15LT AND ( HrLMS(JL'AQUK
TOWNSHIPS. Michael Follmeb, born in Germany in 1723, was the
father of six sons and three daughters, and died in 17'J3, aged
seventy years and three days. He and his sons donated the site of
the Follmer Lutheran church, and a farm of eighty acres to the
congregation, and, with his wife, ho was buried in this church yard.
He left a tract of land near Milton to his sixth son, Henry Follmer, an
active member of the church, who married Susan Stohl, by whom he
had iive children: Philip: Andrew; Thomas; Maria, and Elizabeth. He
died in 1822, aged tifty-four years and nineteen days, and his widow
died in 1861, aged eighty-nine years, eleven months, and nineteen
days. Andrew Follmer, the second son of Henri- Follmer, inherited
sixty-eight acres of land, upon which he reared two sons. Henry P.
and Reuben T. ; the latter is dead. He obtained a common school
education, and was an elder of the Reformed church at the time of
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