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

Sketchpad Handbook

Introduction

If you (or students) are not familiar with Geometer’s Sketchpad, this is the book for you. This book is designed to introduce middle school, high school students, and teachers to the basic tools of Geometer’s Sketchpad. This handbook is a teaching tool. It is intended to supplement, not replace, the school’s geometry curriculum.



During the first or second day of your work sessions, students should carefully read part one, establishing a good study program, and then move on to complete the free play tour. When possible, illustrations are provided to help visualize the various Sketchpad tools and to enhance written instructions. The tours and investigations will encourage you to work together in pairs and small groups to build on previous knowledge and apply knowledge to new information. Students learn well working in groups. When students get stuck on something, they say so, and other students help. This kind of learning is efficient, natural, and well- known.



The Geometer’s Sketchpad, or 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 and CORE Mathematics Curriculum.





Sample Unit in the Sketchpad Handbook

Geometric Transformations

A transformation is a movement of a geometric figure to a new position. One kind of transformation is a reflection.

Reflections using Geometer’s Sketchpad

A key word in connection with reflection is mirror. In order to reflect a selected object in a line, the line must first be selected and marked using Transform | Mark Mirror – refer to Stickman 1 below.

Draw Mr. Stickman on the left of the segment. Reflect Mr. Stickman in a line. Then reflect that reflection in a second line. Now vary the lines. Save your work – refer to Stickman 2 below.

 

Deepening your Understanding of a Reflection

A reflection is sometimes called a flip. The figure is flipped across or over or in a certain line. Observe that triangle ABC is reflected across the y-axis to make ABC. The corresponding points of the two triangles have the same y-coordinates. Their x-coordinates are opposites – refer to Reflection 1 below.

The figure that results from a reflection is called the image of the first figure under a reflection. RSTU is the image of RSTU under a reflection in the line with equation y = 2. Corresponding points of the two figures are the same distance from the line of reflection, but on opposite sides – refer to Reflection 2 below.

Reflections do not change the shape or size of a figure, so the reflection image is congruent ( is the symbol used for congruency) to the original figure (the pre-image). In the examples above, ABC ABC, and RSTU RSTU.

Reflection Notation:

rx = 0(ABC) = ABC. Reflection over the line x = 0 (the y-axis).
ry = 0(RSTU) = RSTU Reflection over the line y = 0 (the x-axis).
Reflection over an axis changes coordinates of points as follows:
x-axis: ry = 0(x,y) = (x,-y)
y-axis: rx = 0(x,y) = (-x,y)

Test Your Understanding of Reflection

1) The diagram shows ABC. If ABC is reflected across the x-axis to produce ABC, what will be the coordinates of Point A? First use the appropriate reflection notation formula to solve the problem, then check your computations by using Sketchpad to reflect ABC across the x-axis – refer to reflection 5 below.

2) The diagram below shows parallelogram P and its image under a certain reflection – refer to reflection 6 below.

Which describes the reflection?
A rx = 1(P) = P
C ry = 2(P) = P

B rx = 2(P) = P
D ry = 4(P) = P

3) The diagram below shows DEF - – refer to reflection 7below.

Without using Sketchpad Reflect command, draw D’E’F’, the result of the reflection described by ry = -1(DEF).

4) Look at triangle PQR on the coordinate grid below. Refer to Rotation 8 below.

PQR will be rotated 180 about the origin to make PQR. What will be the coordinates of the image vertices of the triangle? Remember to first use the rotation notation to confirm your answers by using Sketchpad to perform the indicated rotation.

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