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

Susan Buksbaum

EUCLID A LA MASON

Robert Mason and Geometry

“I want students to see math, to appreciate it, as part of a quest - another way of trying to make sense of the world,” says Senior Math Teacher Robert Mason.  “It’s a body of knowledge important  to understand for its own sake, but it’s about thinking - about the analytical process of solving problems.”  His Euclid Project, taught to both 7th and 8th grade math students, is the culmination of his efforts to integrate technology into math education, to help the non-intuitive math student, and to make what he sees as real connections between math and the humanities.

Mason believes strongly that the early Middle School years must be spent solidifying computational and problem-solving skills.  The 6th grade includes some pre-algebra as well.  With those basics mastered, 7th graders begin the Euclid Project, an informal study of Euclidean geometry which enables students to explore the theorems and definitions of plane geometry without having to write the formal proofs that are a part of high school math. Half of the project is completed in 7th grade; 8th graders complete the rest before moving on to Algebra I.

Responding to a recommendation by the NCTM that more stress on geometry was needed in eary math education,  Mr. Mason saw the opportunity to give students the chance to explore two-dimensional space in an informal, child-centered fashion.  The 7th grade course begins with an investigation of the process of analysis itself, ranging from defining the word to its application in studying the humanities as well as math or science. (His Core colleagues participate in this conversation in their classes at the same time.)

Next, the scientific method is introduced:  hypothesis, testing and conclusion.  Using familiar Euclidean theorems, Mason facilitates the development of hypotheses, offers models of experiments to test them, and guides the independent explorations of students  for findings that will support the hypotheses.  Stressing the cause and effect relationships basic to most learning, he says he “tries to give students the same analytical learning structure that is fundamental to all thinking, whether in Math, Social Studies or English.”

The use of computer technology, he asserts, is a crucial tool. Geometer’s Sketchpad, the software program he uses to support the curriculum, contains “certain tools one can use to build shapes and move them around the computer screen, which enhances the ability to visualize and review one’s thinking.  It provides an environment in which 7th and 8th graders can play with geometric concepts like they did as toddlers, playing with blocks.”  A confirmed constructivist, he says, “You learn best when you are actively participating in learning. “  Computers have made it possible “to create learning situations in which students are first taught about the tools they’ll use and then can use the tools to make meaning of their explorations.”

Then comes the writing.  Students are required to record what they’ve done and to describe their thinking during the process:  “Describe what you did; what are your findings, why does it make sense?”   These “mini-essays” not only help students clarify and verbalize the process they have experienced, but help them to look more broadly at the mathematical concepts involved.  He asks them to “look for relationships between problems that will help them grasp concepts at a high level; “to “take inventory” of a problem, search for familiar elements, and to use information cumulatively, instead of storing it in “discrete chunks.”

The last component of the program is  an application of the knowledge they have mastered. Mason explains that, while they use the same models to show that they understand both the content and concepts of the course, in this case they are “paper and pencil” exercises; in reality, old-fashioned tests.

The program, he explains, not only empowers the students, but “frees the teacher to work on the thinking rather than the mechanics, to communicate more with the students about the thinking process involved in this kind of work.”  He asserts firmly that though the project is dependent on the availability of technology, computers are a tool, not the driving force at work in the curriculum.  Without appropriate software, for example, the eighth grade algebra curriculum relies on  a traditional textbook.  The teacher remains the vital force in the classroom, he maintains; “they need the human guidance only a teacher can give.”

The Euclid Project meshes perfectly with the Dalton Plan.  While Mason incorporates many teaching styles, he thinks of himself as a facilitator of the process of learning for his students.  Because they work in  small groups, many classes become labs where he works with individuals or groups.  Students who have difficulty with this methodology are referred to textbooks and given an alternative method of solving the same problems, though he says “they miss the opportunity to have a set of mistakes from which to learn.” Individualization of the process  plays a significant role.

His excitement about this curriculum is palpable; not only for teaching mathematics, but the broader use of the process that the project entails.  “This program adds a layer of understanding about the world,”  he explains.  What he omits to say is that it clearly will add a generation of able and competent math students.

 

 

 

 

Susan Buksbaum, Editor

Connections Magazine

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