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

Modes of Approaching Math

page 1

I've Learned that Students have Different Modes of Approaching Math

Doc At WorkAt the closing of each academic school year, I ask myself one question.  Have I effectively taught mathematics to students with a broad range of learning styles?  Up until 1977, my conclusion was, usually not.  During the 1977 summer vacation, I was determined to rectify this problem.  That summer, I designed a reading program in the fields of learning and teaching. From these readings and through self-reflection, I realized that my perceptions of the learner, and how these perceptions conditioned my approach to teaching. I had usually taught mathematics through lectures and demonstrations. Given few opportunities to think for themselves, my students learned that they would be rewarded if they sat passively listening to my understanding of mathematics. When unlocking the learner's intellectual potential became my objective, and constructivist teaching became the framework in which I taught, the dynamics of my classroom changed significantly. 

            When I say I seek to “unlock students’ mathematical potential,” I mean that I believe that students come to my class with a store of knowledge--their potential-- and are often unaware that their existing knowledge of mathematics is part of a larger system.  Enabling students to both recognize the knowledge that they possess and relate their knowledge to new information is my goal. 

            My curriculum provides situations in which students experience this recognition and internalize these relationships.   For example, if seventh graders are given the problem, "combine 1/a + 1/b," they know, from arithmetic, how to add positive rational numbers with unlike denominators.  Therefore, teaching students to recognize that adding 1/a + 1/b involves the addition of positive rational numbers is the first step.  The process of unlocking potential, in this case, occurs when students recognize that they can apply some of their knowledge about adding fractions with unlike denominators to combining the algebraic expression problem, and then determine what is needed to bring closure to this problem. 

             By structuring math activities around questions and procedures, my teaching is constructivist. It directs students to survey their knowledge base to locate  “chunks” of information that apply to the problem. They figure out what they understand about a problem, use this information to begin problem solving, and then speculate about what is needed to further their progress.

            When students fail to identify useful information, we brainstorm about mathematical concepts, procedure, and problem-solving strategies.  My role in these brainstorming sessions is to pose questions which initiate general discussions about the next step in their thinking and problem-solving process.  For instance, I will ask, “Is there something in the work we did that points us in the direction of the next step?”  Through modeling, constructivist teaching trains students to initiate these prompts for themselves. When learning at their own rate and viewing mistakes as learning opportunities, students are put in the role of “makers of meaning.”  As a result of my transition to a more encompassing vision of my students, I’ve discovered that students “make meaning” in various ways.  I’ve increased the opportunities for learning by developing several teaching approaches to convey the same information to different learners.  While learning is individualized, students share in the common search for the particular keys to their potential. All students get a taste of these diverse approaches so that their perspective of math and thinking broadens.  

 

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