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
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.

Victoria Geduld

This book opened my mind in a whole new way to thinking about what I do. 

 

Victoria Geduld is a visiting professor at Barnard College and a Lecturer at The New School in New York. She is working on the final stages of her dissertation at Columbia University in the Department of History specializing in 20th-century cultural history and foreign policy. While at Columbia, Geduld served as head Teaching Fellow for Alan Brinkley, Carol Gluck, and Anders Stephanson. She was named Teaching Fellow coordinator and trained incoming student teachers, and will teach at Columbia College in the 2011 summer program.

 

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 both young and experiences teachers in all disciplines. Chapter 1, "Retrospection: My Humble Beginning," takes a new teacher through initial challenges. Although Dr. Mason set up posters for his grade school students, and we establish CourseWorks or Blackboard sites associated with a university-level class, Dr. Mason emphasizes that organization and classroom designs set the tone for productive interchange thus enabling "success" versus "survival" for any teacher; this is true for us all. He reminds new teachers that these routines, including regular and timely assessments and response, are vital for student learning. He establishes the importance of working with others in the institutions who have more experience and understand the politics of the working environment - key advice for anyone in the university setting. Dr. Mason counsels new teachers to review their "constituencies." Although his would be, "parents, teachers, and students," and a TA would consider, "department, supervising professor, and students," the advice remains poignant.

 

Perhaps of greatest importance for new university teachers are Dr. Mason's ruminations on the weeks leading up to the first class; Ph.D. students are wary of admitting trepidation or insecurity. Yet in many and even most cases, Ph.D. candidates have had little experience in the classroom, and do not have post-graduate training in pedagogy. We, too, experience, "anxiety, sleepless nights, and long hours of planning and writing a curriculum." Just as Dr. Mason suggests, we, too, ask ourselves "How do I begin? What should I do first?" Indeed, some Ph.D.s informally advise others that an authoritarian position is mandatory. Others create a formal, yet relaxed atmosphere. Dr. Mason invokes the old adage, "To thine own self be true," when he writes, "all teachers approached the opening day of school differently." Indeed, some PhDs have never hesitated to say, "I don't know the answer, but I'll be sure I get you the answer for the next class." As Mason suggests, any teaching tools that encourage discussion "energize" the student.

 

Chapter 2 continues Dr. Mason's odyssey three years after he began teaching. As Ph.D. students, we only teach for three years. Perhaps his book will enable the acceleration of these early stages. In addition, Ph.D. candidates go on to teach their own syllabus. Dr. Mason suggests that at the more advanced stages we can become comfortable enough to "become more sensitive to students' verbal responses." We can develop the courage to entertain questions during lectures delivered at podiums, thus breaking through the "fourth wall" of the classroom to engage productive learning through interchange. Although Dr. Mason interacts with grade school students of up to 30 in a classroom, and we may lecture to large halls of over a hundred undergraduates, Dr. Mason challenges us to break through the model of viewer and performer that plagues university teaching for less experienced professors. Our most renowned professors manage interchange in the largest lectures with grace and confidence. We will never achieve this without experimenting. Mason asks us to take his dare. Of most import, however, are the three aspects that define his conceptual framework for the classroom: competing for students’ attention, logical thinking encouraged by concrete examples, and finding the zone of “proximal development.”

 

First, the competition for students' attention is an area of academic concern. Many differences remain between the grammar school student and the undergraduate, and I argue that the undergraduate professor has a harder time than Dr. Mason, even with his group of pre-teens and teen-agers. Our bid for student attention includes competition from the internet and Facebook, on-line shopping, and even solitaire. Understandably, some professors have taken a "Just Say No" policy to computer use during classes. However, this does not allow a student to take notes and then coordinate these ruminations into a document that can later be made into a study guide or a list of possible paper topics inspired in the moment and forgotten with the pen in hand. For those who grew up with the keyboard, creative flows often take place while typing. Should we just shut this down rather than engaging the students? Several years ago, internet-based research was also a no-no, but now it is taught as a practice. Dr. Mason suggests that with the right tools, we can make the same breakthrough in the classroom setting.

 

Second, the use of concrete examples to bring forth logical thinking would seem to be a natural outgrowth of teaching the discipline of history. Yet we often get mired in the conceptual frameworks of how to "do" history, and forget to teach with concrete examples. Mason’s pedagogy suggests that we should reference passages from the readings for that day by quoting them and asking students to respond: simple and effective. In the past, I have not done this since I assume that I am building upon these readings. Yet how simple to validate the students' experience by explicitly asking them to integrate preparatory readings into the lecture? In addition, Mason validates the assignment of brief primary documents to supplement secondary texts. These moves inspired by Mason are often overlooked, even in a profession built on documents and "facts."

 

Third, Dr. Mason writes about "a zone of proximal development," or "the difference between the child's capacity to solve problems on his own, and his capacity to solve them with assistance." In essence, he is challenging the university professor to work with students so they can formulate their own ideas and not merely parrot back facts on multiple choice exams. Our function is to define the historiography and present documents necessary to build an argument, but not to dictate an idea, or a series of arguments, in such a way that the student assumes that these, too, are "facts." Dr. Mason suggests that we must give students the historical discipline’s ‘vocabulary’ and ‘grammatical structures’ so that they can create their own sentences and new ideas.

 

In Dr. Mason's chapter, "Resolution," he sums up earlier points and allows the reader to understand their implications. For example, we should allow silence to linger when it is productive, and intervene when it seems frustrated. New teachers generally understand silence as failure, particularly when the teacher asks a question. Dr. Mason encourages us to both allow students to think in silence, or understand when a question needs to be explained due to the qualitative behaviors exhibited by the students. Second, reviewing is not a crime. We also tend to view the lack of student recall as a reflection of our earlier efforts or student disengagement. Dr. Mason suggests that perhaps the concept was forgotten, misunderstood, or even presented just before vacation, a large paper due-date, or an exam period when students are distracted. Whatever the reasons, he suggests that we must not merely push on. Without the review of key concepts, students will become even more muddled, and, in turn, shut down as active participants in the classroom exchange. Reviewing is an art.

 

Although Dr. Mason writes about teaching mathematics to grammar and middle school students, his reflections and revelations should be shared with undergraduate history teachers who will benefit from this book as much as a mathematics teacher. When conducting the New Teachers Seminar for incoming Teaching Fellows at Columbia, I only wish that I had known about Dr. Mason's book; I would have assigned it immediately and used it as the basis for the first discussions. As I move forward with my career in history, when asked to mentor teachers again I will certainly assign Dr. Mason's Teaching Mathematics.

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