Category Archives: Education

Plane and Solid Geometry

These sections are taken from PLANE AND SOLID GEOMETRY by George Wentworth and David Eugene Smith, first published in 1888.

G00 – Contents

G01 – Introduction

G02 – BOOK I. Rectilinear Figures

G03 – BOOK II. The Circle

G04 – BOOK III. Proportion. Similar Polygon

G05 – BOOK IV. Area of Polygons

G06 – BOOK V. Regular Polygons and Circles

G07 – Appendix to Plane Geometry

G08 – BOOK VI. Lines and Planes in Space

G09 – BOOK VII. Polyhedrons, Cylinders and Cones

G10 – BOOK VIII. The Sphere

G11 – Appendix to Solid Geometry

G12 – Miscellaneous & Index


Troubleshooting Math

To troubleshoot any difficulty you first look at the broad area of that difficulty, and then you gradually narrow it down until you have defined the actual difficulty precisely.

So, to troubleshoot a difficulty in math you start with the broad area of Mathematics.

Mathema (Greek) = Learn
Mathematics = Tools for learning

Mathematics provides you with analytical tools for learning. When you are troubleshooting mathematics, you are troubleshooting the difficulties a person is having with learning analytically. You narrow down to the area of mathematics where the person cannot think analytically.

Mathematics is analytical learning and not just memorizing of materials.

If the student is having trouble with higher mathematics, such as, Trignometry, Analytical Geometry, or Calculus, then start from there. You may explain the area the student does not understand. But if the student cannot understand the explanation analytically, then the troubleshooting may lead to one of the three areas below.

When you select one of these areas, explain it per Math Overview. You do not have to explain that whole document. Keep to the trail of trouble.

Ask, “What part of this area you have most difficulty with?”

Use the answer to narrow down further to the area of difficulty. Quiz the student on the key math vocabulary in that area. From student’s answers you may narrow down the area of difficulty further.

If the student cannot answer the question, simply start with the first lesson
related to that area at Mathematics. Follow student’s attention to fish around for the actual difficulty.

As you narrow down the area of difficulty, keep asking, “What part of this area you have most difficulty with?”

Check the key math vocabulary in the narrowed down area. Soon you’ll reach the actual difficulty. Handle it using the right materials selected from the appropriate level at Mathematics, or from student’s own materials.

Once that area is handled, the student may come up with another area that he or she has attention on. Narrow down to the actual difficulty in that area as above, and handle it.

Otherwise, start all over again from the diagram above. This time you may follow a different trail to a different area of difficulty.

Ultimately, teach the student how to troubleshhoot difficulties. This is the best thing you can ever do for the student.


Stress and Education


Reference: Critical Thinking in Education


The biggest challenge to education is the stressed child, or the stressed student. When a child is stressed his attention is introverted onto his personal issues and he cannot learn.

The education at SLS is successful because it is addressing the challenge of stress successfully through its special curriculum. Learning requires extroverted attention.  The SLS environment is very extroverting.

Rule: The school environment should be such that it extroverts attention.

The general stress in the current society is increasing. It is inevitable that a certain percentage of children coming to school have stressful situations that are holding their attention. Their introverted attention then does not allow them to learn.

It is absolutely necessary for school to provide a stress-free extroverting environment so that learning can take place. If the school’s environment is also stressful then the student becomes conditioned and robotic.

At SLS, the first half hour of the day is devoted to activities that extroverts attention. The following exercise may also be used to extrovert attention.

This exercise may be conducted with a group of students, or it could be applied to a student who has difficulty learning.



PURPOSE: To extrovert the attention by exploring the five physical senses.


(Touch – 5 minutes minimum)

  1. Go to an environment where you can explore the sense of touch.

(a)  Touch two different surfaces and compare how they feel.

(b)  Touch them alternately until you can discern the uniqueness of each surface.

(c)  Touch a third surface repeatedly to get a feel of it. Then touch it alternately with one of the earlier surfaces, until you can discern how this third surface is unique.

(d)  Similarly touch additional surfaces carefully until you can discern their uniqueness.

  1. Explore the sensation of touch until you can do so happily without feeling any resistance inside you.

  2. Exercise the sense of touch for at least 5 minutes. You may do it for as long as you want.


 (Sight – 5 minutes minimum)

  1. Go to an environment where you can explore the shapes and colors of things.

(a)  Look at two different objects and compare their shapes and colors.

(b)  Look at them alternately until you can discern the uniqueness of their shapes and colors.

(c)  Look at a third object repeatedly to get an idea of its shape and color. Then look at it alternately with one of the earlier objects, until you can discern how this third object is unique.

(d)  Similarly look at additional objects carefully until you can discern their unique shapes and colors.

  1. Explore the sight of objects until you can do so happily without feeling any resistance inside you.

  2. Exercise the sense of sight for at least 5 minutes. You may do it for as long as you want.


 (Hearing, Smell & Taste – total 10 minutes minimum)

  1. Sit around a table and unpack your lunches and drinks. Don’t hold yourself back from talking.

  2. Start smelling and tasting little bits of your lunch, while listening to each other talk. You may even listen to your own voice.

(a)  Explore the different sounds that you hear as to their timbre, pitch, loudness and other qualities.

(b)  Explore the different odors as to how pleasant or pungent they are, and as to their other qualities.

(c)  Explore the different tastes as to how sweet or salty they are, and as to their other qualities.

  1. Explore the sounds, smells and tastes until you can do so happily without feeling any resistance inside you.

  2. Do this exploration for at least 10 minutes. You may do it for as long as you want.

  3. Take some deep breaths, appreciate what is around you, and get ready for your next school activity.


The SLS Math Course


Reference: Critical Thinking in Education



The SLS Math curriculum is designed with the following rule in mind.

RULE # 1: The curriculum follows the sequence in which concepts are developed systematically in a subject.

The subject of mathematics starts with COUNTING. The next concept is PLACE VALUE. Place values allow one to write large numbers in a concise manner. The student must learn how to read and write large numbers before proceeding to the next concept of ADDITION.

Mathematics introduces the student to systematic learning. Counting and place values provide ways to think systematically.



The SLS Math curriculum consists of lesson plans that are concise, relevant and easy to follow. The students are encouraged to read and understand the lessons on their own. Supervisors are there to help him as needed.

RULE # 2: The lesson plans are concise, relevant, and written in plain language that is easy to follow.

Each math lesson is followed by a large number of exercises for practice. Answers are provided for all exercise problems. The students are encouraged to do the exercises and check their answers. The correct answers reinforce the students’ confidence.

RULE # 3: Each lesson plan is followed by a large number of exercises, with answers provided for all exercise problems.

The students are encouraged to trace the incorrect answers back to the exact error made.  Supervisors are there to assist them in this effort. Once a student becomes aware of the exact error he is less likely to make it again.

The student works to get the correct answers first, and then works on the speed. He learns the methods of arithmetic that make computations easier and faster.

The student may do every fifth or every tenth problem first to sample problems of different level of difficulties. He may then practice the problems that are at the right level of difficulty for him..



When the student has studied and practiced a lesson plan he asks the supervisor to check him. The supervisor spot checks him on the concepts of the lesson and have him solve some exercise problems. If the student fails the spot-check the supervisor sends him back to study and practice some more, and come back for another spot-check. When the student passes the spot-check he goes to the class tutor to be examined on his understanding of the lesson plan.

The class tutor examines the student’s knowledge from the viewpoint of skill. He makes sure that the student has required skills. If the tutor finds some minor things missing in the student’s understanding then he tutors him on the spot. If he finds something major missing then he sends the student back to the supervisor with exact instructions on what the student must restudy and practice.

In the end, the class tutor requires the student to do three exercise problems correctly in a row. When the student answers all three problems correctly, the class tutor announces him complete on the lesson plan.

RULE # 4: In order to complete a lesson plan, the student must solve three exercise problems (of reasonable difficulty) correctly in a row.



Classes are divided by the levels of the curriculum. Levels Pre-0 and 0 are written for skill levels learned in Pre-kindergarten and Kindergarten respectively. Similarly, Levels 1 and 2 are written for skills learned in primary and middle school respectively. Each level consists of a number of lesson plans. When a student has completed all lesson plans for a level, he moves up to the next level.

If the student is found lacking the skills of a level he is assigned to that level. He is then examined for completion of each lesson plan on that level.

The SLS math course is performance based. The students can move through these levels rapidly. He is not held back because of age. Normally a student is allowed to advance through these levels at a pace most suitable for him. By the time a student has completed Level 2 he is deemed to be a self-learner. He then continues up through Level 3 and above rapidly with minimal supervision..

A higher level student is also trained on supervisor skills. He supervises at least one lower level student through to completion.

RULE # 5: A higher level student must be able to assist a lower level student to completion.


Arts and Maths

Arts and Maths

Math and Arts are two very different subjects that lie at the opposite ends of a spectrum of “Expression of Life”. We express life in many different ways to explore and understand all different aspects of it. In my view, Arts is much more complex than Math. What is being expressed through music, theater and drama is millions of times more complicated than the expression of simple rules through arithmetic, algebra and geometry.

Mathematics is like trying to understand the dimensions of Arts in its atomic simplicity. A person who has mastered mathematics can truly appreciate Arts to its deepest dimensions. What an artist does intuitively, he can do it with much greater understanding and mastery, once he has understood mathematics.

We talk about being left-brained or right-brained… we talk about being analytical or intuitive… but these “opposites” are part of the same spectrum of understanding. Therefore, it is very possible that we can be both left and right brained, or both analytical and intuitive.

If you are an artist, just check out mathematics. You have some revelations coming your way.