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

An Example of my Approach

 

 

The Apple Lesson - Making the familiar unfamiliar

 

 

I had just begun to tell the following story.

 

Apple1“Sometimes it is a useful endeavor to look at familiar objects or ideas in unfamiliar ways. Moreover, the reverse is also true. That is, it is sometimes helpful to look at unfamiliar objects or ideas in familiar ways.  Today we will analyze an apple in new ways, and then discuss the undressing of an apple.  Before we undress our hypothetical apple, let’s talk about its components. The first thing we notice is that our apple has an outer component that we call the skin.  The skin encloses the interior, or inside, of the apple.  As the apple matures and grows the skin expands mostly in surface area. The increase in the amount of skin over a period of time is an accumulation of matter and the effect of this accumulation is similar what we call addition in mathematics. 

         At the same time, we find the pulp underneath the skin in volume as the apple matures. The natural growth cycle of the pulp is similar to a layering effect.  Consequently, the expansion in volume is also an accumulative process.  It is then possible for one to view the repeated layering as equivalent to repeated addition, and if repeated addition is equivalent to multiplication then repeated layering is the same as multiplication.. The symbolic depiction below represents our apple equation.

Apple = pulp x core + skin

         Now, let us undress the apple in a physical sense and see how the process can be paralleled by manipulating the symbolic form.

 

Apple - skin = pulp x core + skin - skin 

Pulp (t = 2)=pulp (t = 1) x layer

Apple2skin (t = 2) = skin (t = 1) + add’tl skin

core(t = 2) = core (t =1)

 

As we evaluate our apple and apple equation, we notice that what remains is pulp and core on both sides.  Our next step is to remove the pulp. One physical method is placing ones thumbs on top of the apple where the stem is located and the fingers around the bottom of the apple, one then applies pressure with the thumbs and pulls the apple a part.  The motion is similar to an inverting.  The mathematical equivalent to turning something upside down is referred to as the reciprocal, or inverse. If we agree to let pulp/1 represent the symbolic form of pulp, then the reciprocal of pulp/ 1 is 1/pulp. 

         To remove the pulp from the actual apple means to separate the portions that have been designated as pulp from the core.  Any method one uses to remove the pulp form the core must preserve the core.  Therefore, all methods used would be similar because the pulp would be removed by a “taking away” gesture which separated small portions of the pulp from the core.  If the removal of the pulp is done by a repeated “taking away”, and repeated taking away is equivalent to the mathematical operations of repeated subtraction, then to remove the pulp from the symbolic apple one divides both sides by M/1 or multiply both sides by 1/M.  Where 1/M refers to the portions used to remove the entire pulp. 

 

1/ pulp x pulp /1 x core = Apple - skin x 1/ pulp

                        or

pulp x core/ pulp =  Apple - skin/ pulp

core = Apple - skin/pulp

Now let’s use the above process to a specific apple equation. 

12Apple = 5pulp x Core + 2skin

We first remove the skin from both sides.

12Apple  -  2skin  = 5pulp x Core + 2skin - 2skin

We are left with the following apple equation.

 

 Apple3

 

10Apple - skin =  5pulp x Core

 

To remove the pulp we divide both sides by the pulp.

10Apple - skin/ 5pulp =  5pulp x Core /5pulp

2Apple - skin/ pulp =  Core

 

 

 

 

 

 

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