Hi, I'm Robert!
For my fellow teachers, I've constructed a model of teaching that I've summarized as the puncturing of space with pedagogical objects. . . The term "objects which puncture space" may help solidified one's sense of how pedagogy can be described within its new conceptual framework. Teachers who see the world in this manner should become more fully invested in the enterprise of teaching and learning.
Teaching Methematics
S.T.O.R.E.S.
for teachers
S.T.O.R.E.S.
for students
Handbook
The Euclid Project
Teacher's Manual
The Euclid Project
Student's Manual
An Introduction
to Geometer's Sketchpad
The Euclid Project
Pre-Algebra
Teaching Mathematics
"Teaching Mathematics Puncturing Space: A Developing Pedagogical Tool" uses a diverse
body of research to clearly introduce important ideas related to learning. Theories from
the fields of neurology and cognitive development about how students obtain, synthesize
and retain information are examined and cohesively presented.

With an in-depth discussion of how educators compete with predictable outside stimuli
as well as with the internal life of the student mind, Dr. Mason explains the idea of
using a combination of objects as pedagogical tools to 'puncture' the learning space to
re-engage the student and to re-establish attentive behavior.

This readable book is valuable to educators in all fields not just to those teaching
Mathematics, and not just to those teaching in lower and secondary schools. Educators
will think carefully and differently about how information is delivered and processed
in the classroom, after reading this book.
S.T.O.R.E.S.
(for teachers)
Structured Teaching of Research and Experimentation
Skills (S.T.O.R.E.S.) science curriculum for elementary
school and middle school students is a process oriented
approach, focusing on classical principles of induction
and deduction, evidence gathering, and hypothesis
building, and empirical testing and refinement of
hypotheses that highlights scientific procedures.
S.T.O.R.E.S.
(for students)
Structured Teaching of Research and Experimentation
Skills (S.T.O.R.E.S.) science curriculum for elementary
school and middle school students is a process oriented
approach, focusing on classical principles of induction
and deduction, evidence gathering, and hypothesis
building, and empirical testing and refinement of
hypotheses that highlights scientific procedures.
Sketchpad Basics
Handbook
Sketchpad Basics Handbook is designed to introduce elementary school and middle school students
and teacher to Geometer’s Sketchpad. The Sketchpad, is a construction tablet on which one draws models of geometric shapes, transforms them, colors them, measures them, and animates them. The models invite students to explore, represent, solve problems, construct, discuss, investigate, describe, and predict. Implicit to these functions is the ability to build mathematical models of simple and complex ideas. The Sketchpad allows students to engage in “doing mathematics,” which is emphasized in the National Council of Teachers of Mathematics (NCTM) Standards.

The investigations encourage students to work together in pairs and small groups, and to build on their knowledge by applying their knowledge to new information.

Sketchpad introduced through a series of explorations. All of the explorations are designed specifically to teach how to use the “tool box.” They represent technical exercises. That is, they teach how to use the drawing tools, and how to use the command menus to accomplish specific task. In some investigations students will replicate as set of instructions and then evaluate their findings. In other activities students are free to create their own investigation.
The Euclid Project
Teacher's Manual
The Euclid Project computer-based geometry program uses a scientific-experimentation approach to
providing middle school students with an intuitive un?derstanding of geometry as a precursor to the formal study of geometry later (e.g., in the 10th grade) and as a mediator for application of geometric understanding in a variety of contexts.

This scientific-experimentation approach to teaching geometry involves pre?senting the students with a mathematical hypothesis
(e.g., a line drawn across two sides of a triangle parallel to the third side divides the first two sides proportionally),
then having them use a “construction tablet” (Logo, Geometer Supposer, Geometer’s Sketchpad computer programs) to systematically
generate a series of cases to test the validity of the hypothesis (e.g., create a triangle and line parallel to a side,
then use animation to gener?ate a series of such triangles to see if the hypothesis holds for all of them).
The Euclid Project
Student's Manual
The Euclid Project computer-based geometry program uses a scientific-experimentation approach to
providing middle school students with an intuitive un?derstanding of geometry as a precursor to the formal study of geometry later (e.g., in the 10th grade) and as a mediator for application of geometric understanding in a variety of contexts.

This scientific-experimentation approach to teaching geometry involves pre?senting the students with a mathematical hypothesis
(e.g., a line drawn across two sides of a triangle parallel to the third side divides the first two sides proportionally),
then having them use a “construction tablet” (Logo, Geometer Supposer, Geometer’s Sketchpad computer programs) to systematically
generate a series of cases to test the validity of the hypothesis (e.g., create a triangle and line parallel to a side,
then use animation to gener?ate a series of such triangles to see if the hypothesis holds for all of them).
An Introduction to
Geometer's Sketchpad
This workbook is designed to introduce elementary school and middle school teachers to Geometer’s Sketchpad.

The Sketchpad, is a construction tablet on which one draws models of geometric shapes, transforms them, colors them, measures them, and animates them. The models invite students to explore, represent, solve problems, construct, discuss, investigate, describe, and predict.

Implicit to these functions is the ability to build mathematical models of simple and complex ideas.
The Sketchpad allows students to engage in “doing mathematics,” which is emphasized in the National Council of Teachers of Mathematics (NCTM) Standards.
The Euclid
Pre-Algebra
description

Teaching Mathematics - Introduction

Introduction



Those who can, do. Those who can’t, teach.



We have all heard this statement expressed many times by intelligent individuals, individuals who themselves are successful in fields other than teaching. I find the sentiment most insulting and misleading-- insulting because it demeans and devalues the efforts and achievements of past and present teachers, scholars, philosophers, and theorists in the field of education, and misleading because it carries the implication that intelligent people choose not to teach because teaching does not require intelligent “doing.” Furthermore, the statement can only be based upon the experience of the speaker, and that experience is, of necessity, based only on personal experiences and interactions with a small and particular set of teachers. The sentiment expressed bears only on the substantive personal knowledge those teachers possess. Yet, there are broader intellectual components of teaching that are often overlooked or ignored because they are rarely discussed with students or, indeed, with those not intimately involved in public or private education. For instance, many teachers spend countless hours thinking about the learner and activities that promote “learning” in an effort to determine how to best, or most effectively, stimulate the mind of students to store, retrieve, analyze, synthesize, and evaluate information.



The first step in intelligent doing occurs when teachers analyze and assess various theories of learning. As teachers sort through competing theories, they then spend years translating educational theories. Some then spend years translating these theories into practical applications that seek to “unlock the intellectual potential of the students.” When I say we seek to “unlock students’ mathematical potential,” I mean that students come to a class with a store of knowledge and their potential-- yet are often unaware that they possess this knowledge or that their existing knowledge of mathematics is part of a larger system. Enabling students both to recognize the knowledge that they possess and to relate that knowledge to new information should be the goal of a curriculum. An effective curriculum should provide situations in which students experience this recognition and internalize these relationships. The ones I describe in this book provides a working example. I call this kind of thinking and curriculum development – “intelligent doing.”



I think that it is fair to assume that most non-teachers would at least acknowledge the fact that teaching is done in a social context that is complex. And, they would agree that the intellectual potential that teachers strive to unlock is sometimes easily accessible and sometimes not. Moreover, most would agree that some students possess intellectual abilities that require a qualitatively different kind of “unlocking” than those whose potential is not easily accessed.

How, when, and whether teachers get to the stage in their teaching where they actually think about teaching and learning is still not well understood. I believe that some teachers go through what I call a pedagogical transformation that occurs as a result of direct as well as indirect interactions with the cultural framework of their schools, that is, with the implicit and explicit values, traditions, beliefs, patterns of behavior, and assumptions that are held by the various constituencies of the community. It is within this context that teachers develop their approaches to teaching and determine the degree to which they satisfied, or dissatisfied, with their pedagogical, or instructional practices. It is within this context that the “doing” of teachers is intelligent.



I think that it is fair to assume that most non-teachers would at least acknowledge the fact that teaching is done in a social context that is complex. And, they would agree that the intellectual potential that teachers strive to unlock is sometimes easily accessible and sometimes not. Moreover, most would agree that some students possess intellectual abilities that require a qualitatively different kind of “unlocking” than those whose potential is not easily accessed.



How, when, and whether teachers get to the stage in their teaching where they actually think about teaching and learning is still not well understood. I believe that some teachers go through what I call a pedagogical transformation that occurs as a result of direct as well as indirect interactions with the cultural framework of their schools, that is, with the implicit and explicit values, traditions, beliefs, patterns of behavior, and assumptions that are held by the various constituencies of the community. It is within this context that teachers develop their approaches to teaching and determine the degree to which they satisfied, or dissatisfied, with their pedagogical, or instructional practices. It is within this context that the “doing” of teachers is intelligent.



Part of the problem lies in the fact that teachers’ have failed to share the details of their intelligent doing with a larger audience. And, because of this, non-teachers have formed an opinion of teaching that is based on incomplete information and thus lacks an appreciation of the depth and complexities of our work. Their opinion is “boundedly rational” – rational because it is based on the experience of at least twelve years of formal education, and bounded because the twelve years constitutes but a small sampling from the total population of K-12 teachers in the country. Let’s think about what I’ve said. When we are asked what we do for a living, teachers generally reply by naming the subject they teach and to whom – end of conversation. For the most part we don’t discuss the details of our intelligent doing or our efforts to unlock the intellectual capacities of our students. If we are to influence the opinions of those outside our profession it is essential that we communicate the scope and extent of our efforts to the larger population so that we can bridge the gap between what non-teachers think they “know” about our work and what we actually do. One way this can be accomplished is for teachers to begin to contribute publicly to intellectual explorations, inquiries of the theory and practice through action research.



For instance, classroom teachers can and should participate in general discussions about theory and practice in a manner similar to the way a professional race car driver relates to their teams of expert mechanics. The experts attempt to design and create innovations that will improve the performance of the racing car. Professional drivers in turn gives feedback in terms of their interpretations, the meaning that they attach to their interpretations, and the evidence of how they both effect the performance of the automobile. The experts then evaluate their theories and designs in light of this feedback and, when appropriate, modify their theories. The cycle is clear and effective for automobile racing. But, while teachers are often asked to adopt new theories of learning and pedagogical practices, only infrequently does input from them flow back to the original sources in the form of an articulated analysis and synthesis of the theories. Such syntheses, if they were developed, could be in the form of what Jack Whitehead calls, ‘living educational theories (LET)’ of teachers’ (Whitehead 1984).



If teachers were to participle in the process (much the way racing car drivers do) they would be, in principle, researchers in a practice context. They would not merely apply established theory and techniques but would be central figures in the analysis and assessment of the theory and technique. And their involvement would feed back to the developers of the theory and technique so that the original ideas could be altered and adjusted by incorporating the experience of actual practice.



That is, when teachers reflect-in-action, he/she becomes a researcher in the practice context. He/she is not dependent on the categories of established theory and technique, but constructs a new theory of the unique case. Living educational theories explain events by setting forth propositions from which these events may be inferred, a predictive theory sets forth propositions from which inferences about future events may be made, and a theory of control describes the conditions under which events of a certain kind may be made to occur. In each case, the theory has an 'if...then....' form" (Schon (1983). These contributions would give the original sources insight into their work in terms of how to interpret them and use them in practice.



The importance of teachers' stories of practice as examples of their intelligent doing.



Cochrane- Smith &Lytle (1990), and Carter (1993) contends that teachers' stories of practice are largely an untapped source of information about teaching. Group effort has the potential to help teachers individually as well as the profession as a whole by drawing attention to teachers' stories of practice. They also make a compelling case for why teachers should engage in group inquiries into teaching, stating that missing from the knowledge base for teaching are the voices of teachers themselves, questions teachers ask, and the interpretive frames teachers use to understand and improve their own classroom practice. Florio-Ruane (1991) notes in his research the traditional forms of educational research are largely done by university professors and often does not capture the perspectives and experiences of classroom teacher.



Polkinghorne (1988) has theorized that humans tell stories in order to make meaning from experience. His research into what kind of knowledge teachers use in practice shows that teachers use stories or "narrative ways of knowing". Closely related to Polkinghorne's research is the work of Connelly and Clandinin (1990). They suggest the stories teachers tell about their work are a key means of gaining insights into what teachers know about teaching and how they come to know. Connelly and Clandinin's extensive collaborative research with teachers (Clandinin 1985; Connelly & Clandinin, 1988; Clandinin, Davies, Hogan & Kennard, 1993; Clandinin and Connelly, 1995) indicates that these stories about practice tend to benefit others, and equally important to observe is that, teachers themselves learn in the process of telling, retelling and interpreting their stories. Elbaz (1983) concluded from his research that teachers' knowledge is personal, embodied, narrative, autobiographical and relational. These researchers stress the importance for teachers of understanding their own teaching narratives - their narrative ways of knowing.



Witherell & Noddings, (1991) are strong advocates for recognition of the importance of narrative and dialogue in education. The stories that teachers tell about their own practice are examples of their ‘intelligent doing’ because their voices add to the research base on teaching.. The stories teachers tell do not reveal teaching and learning as straightforward but as involving a complex interaction with students and content. Coles'(1989) and Paley's (1979) published accounts of their teaching provide further insight into the use of story as a teaching strategy and the importance of story in their teaching lives.



There is a mounting support for, and an acknowledgement of the significances of what classroom teachers have to tell about their practices in the research arena. In time, the stories that teachers tell about their practice will improve and will be viewed as a valuable and useful sources of information.



Puncturing of Space: A Developing Pedagogical Tool is one story of my personal journey in search of a practical link between my understanding of the Piagetian stages of cognitive development, Vygotskys’ social constructivist theory of mediated learning through cultural artifacts, and my practices of teaching mathematics to students. While teaching has intense idiosyncratic aspects, it also has general applicable aspects. This book can’t transmit the idiosyncratic aspects of Robert Emmett Mason’s experience, but it can illuminate how one teacher went about developing his pedagogy. This example may prove useful to others as they undergo their path to “intellectual doing.”



I view teaching as a continuous competition between the internal stimuli of students and the external stimuli that would attract their attention. I will show how the act of teaching can be viewed as puncturing space with pedagogical objects, though this will take some extended explanations.



I will then propose the following conjecture: Pedagogical space is punctured with objects for the sole purpose of winning the competition for the attention of students that is ongoing between the internal stimuli of the learner and external stimuli from the environment. Such a practical conceptualization of the classical environment in which teaching occurs provides a framework in which teachers can think about their pedagogical practices, what they view as their role as the teacher, and what they view as the role of the student.



In my experience, a bridge can be formed between an understanding of theories of learning and the practice of teaching through the general notion of what I call functional closure, or the tendency of human being to “bound” several aspects of their psychological and physiological realities.



Throughout this manuscript, I emphasize the theme that teachers can make significant contributions to the field of education in intelligent and creative ways. Because I believe strongly that this is not merely a possibility but, in fact, an achievable goal, I illustrate this point with a detailed discussion of my own transformation from an instructionalist teacher into a constructivist educator.



This book itself seeks to be an example of what I’ve discussed above. It is my contribution to the field of education in the form of an analysis and synthesis of ideas that I have been taught in graduate school, gained by personal reading, and developed through experience teaching mathematics to elementary and middle school students. It is an example of ‘the kind of intelligent doing’ that is possible from classroom teachers.



Because, my ideas evolved from, and are the results of, a search for understanding illuminated by my practice in teaching math, the story begins with a description of my thoughts and concerns during the beginning years of my career as an elementary school teacher.

  • By Michael Sturm

    Robert Mason, affectionately known as Doc by both faculty and students, alike, has taught middle school math at Dalton for the last 20 years. ...

  • Frank A. Moretti, Ph.D

    There are times when a rare person, for mysterious reasons, transcends this set of circumstances and feels the inner necessity to locate practice in the context of theory. Dr. Robert Emmett Mason IV, however, has taken on the challenge of integrating his range of experience in a way ...

  • Kenneth Offit

    The Puncturing of Space: a Developing Pedagogical Tool by Dr. Robert Emmett Mason IV, does not fit an easy description. It is part authoritative teaching handbook, part textbook, and part philosophical discourse from a master pedagogue with thirty years teaching experience ...

  • Victoria Geduld

    Dr Robert E. Mason's Teaching Mathematics might seem far removed from productive pedagogical reading that would be assigned to an incoming Ph.D. teaching assistant in a History department. Indeed, this book should be mandatory for teachers in all disciplines at both the beginning and more advanced levels. ...