Online Vedic Mathematics Conference

12th - 13th March 2016

Day 2: Vedic Mathematics Research

Outlines of Research Papers sent to the Conference organisers
These papers will be available to conference participants prior to the conference

1. A NOVEL APPROACH OF MULTIPLYING THREE NUMBERS NEARING DIFFERENT BASES - Jatinder Kaur and Pavitdeep Singh

2. SINGLE DIGIT ADDITIONS - Saee Vitthal Pallavi Patil (III std.)
Documented by Pallavi Patil, Pune, India.

3. Starring at the 9 - Giuliano Mandotti

4. Improvements in Final Exam Results after teaching 11th and 12th Grade Mathematics using a combination of Vedic Mathematics and Visual Aids
- Alex Hankey, Vasant V Shastri and Bhawna Sharma

5. Evaluation of Trigonometric Functions and Their Inverses - Kenneth Williams

6. The sutras of VM in Geometry - James Glover

7. EFFICACY OF VEDIC MATHEMATICS AND YOGIC BREATHING
IN SCHOOL CHILDREN - A PILOT RCT STUDY - Vasant V. Shastri, Alex Hankey, Bhawna Sharma, Sanjib Patra

8. THREE BOOKS AND A PAPER: A GEOMETRY PROJECT - Andrew Nicholas

9. Swami Tirtha's Crowning Gem - Kenneth Williams

10. Teaching Calculus - Kenneth Williams

11. A pathway for people with perceived learning difficulties to embrace and succeed with studies in Vedic Mathematics. - Vera Stevens

Note these last three papers will be available here but the author's presentation will not be given at this conference.

A NOVEL APPROACH OF MULTIPLYING THREE NUMBERS NEARING DIFFERENT BASES
Jatinder Kaur and Pavitdeep Singh
Abstract
Multiplication is one of the most fundamental operation in mathematics. Conventional method of performing multiplication is quite cumbersome and increases with complexity as the numbers participating in the calculation rises.
Vedic maths is one such science which permits to think for different ways to solve a mathematical problem. Currently, there are few techniques for multiplication of three numbers eg. Product of three numbers near a base, general method for three numbers.
In this paper, we present a novel approach to perform multiplication of three numbers close to different bases (eg. multiplying 205*694*893). The technique makes use of the algebraic equation in solving three numbers near different bases which can be considered as a special case for multiplication. Additionally, the technique can be applied successfully on three number near different bases (eg. 34*51*69 or 2006*5993*7998). Different examples will be provided along with proof of the derived approach to facilitate people in learning the new approach.

Starring a the 9
Giuliano Mandotti
Vedic mathematics has one of the most powerful function needed by a student and a person in general: the flexibility to develop strategy skills in problem solving.
There are some operation that we can have to face in our every day life in which we can use an undiscovered power of the wonderful magic number 9.
When you have to face operations like 21-12, or 53-35 and so on the common approach is to follow the conventional rules of subtraction,  sometimes is simple, sometimes not.
But if we look at this operation in a Vedic Maths framework point of view you'll try to solve in a more efficient, simple and mental way: starring at the 9. Extract the 9 and multiply it by the unsigned subtraction from the 2 operands digits and you obtain the correct result!
A quick example: 53-35 = 9*(5-3) = 18.
This paper provides my simple discovery on this VM strategy, with examples showing the properties with a first level of detail, and some other level of detail to where i came , 3-number subtraction, sum, multiplication, division, special cases, connection with 11-multiplication and other sutras, till ending with some question to open your mind.

Improvements in Final Exam Results after teaching 11th and 12th Grade Mathematics using a combination of Vedic Mathematics and Visual Aids
Vasant V Shastri, Alex Hankey and Bhawna Sharma
ABSTRACT:
Background: Many countries wish to avoid problems associated with large working populations in manufacturing industries by becoming knowledge based societies, where GDP is generated by such means as IT and technical innovation. Good grades in mathematics are required for training in sciences, engineering and business. Colleges teaching 11th and 12th grades must train students to obtain top maths grades for them to enter preferred training programs. Here we present 2010-2015 final exam results at a college in Karnataka. 2014 and 2015 final year classes received Vedic Mathematics training and visual aids teaching, e.g. Geogebra and PowerPoint as cognitive tools in courses on algebra, trigonometry, co-ordinate geometry, vectors, 3D geometry and calculus.
Methods: School examination results were inspected; years 2010 to 2013 without use of Vedic Mathematics and Visual Aids were compared with years 2014 and 2015 where those methods had been employed.
Statistical Analysis: used SPSS 19.0
Results: 2014 and 2015 grades improved significantly on years 2010-2013, +17.05%, p < 0.001.
Discussion: This is the first study to estimate quantitative effects of using Vedic Mathematics, despite it having long been in wide-spread use around the world. Previous informal accounts of its methods’ teaching successes have been substantiate demonstrating that their use together with visual aids may significantly improve grades on career-oriented professional examinations. Such methods may be an imperative in all countries wishing to become knowledge based societies.

Geometry – An Holistic Approach
James Glover
During the last fifty years or so geometry in education has diminished both in quantity and in quality and yet the pedagogic use and applications remain as important now as they were then. A new and holistic set of principles provides a simple approach to the fundamentals as well as to the vast array of applications. Since the sutras of Vedic Maths apply to all areas of mathematics it seems reasonable to suppose that they also apply to geometry. One of the hallmarks of the sutras is that they include the human experience. One example is Vilokanam, By Observation. Accepting both the mathematical and psychological nature of the sutras leads to a new orientation within geometry. This paper looks at nine sutras which provide general principles. Each sutra expresses a simple concept that has a broad range of applications.

Evaluation of Trigonometric Functions and Their Inverses
Kenneth Williams
Abstract
This paper shows that finding trig functions without a calculator is entirely possible especially when high accuracy is not needed. For educational reasons we focus here on the case where the angle is in degrees rather than radians. Through the use of a simple diagram and a special number it is shown that we can estimate the cosine of an angle with ease and that this process is reversible. Any required accuracy can be obtained for the cosine and other functions can be found from this cosine value. This means that teaching evaluation of these functions in the classroom is entirely possible.

EFFICACY OF VEDIC MATHEMATICS AND YOGIC BREATHING
IN SCHOOL CHILDREN - A PILOT RCT STUDY
Vasant V. Shastri, Alex Hankey, Bhawna Sharma, Sanjib Patra
ABSTRACT
We report an RCT study comparing effects of Vedic Mathematics and Yogic Breathing practices on working memory, math-anxiety and cognitive flexibility in school children.
Methods
Subjects: 40 higher secondary (8th-10th grade) residential students at Sai Angels School, Chikmagalur, Karnataka, India, were randomly assigned to Vedic Mathematics, Yogic Breathing, and Jogging groups, containing 14, 13 and 13 children, respectively.
Inclusion / Exclusion Criteria: School children who study Mathematics in their curriculum and having normal vision or corrected to normal vision were included. Children undergoing any psychiatric treatment; having neurological disorders and colour blindness were excluded.
Intervention: Children in Yoga Breathing and Vedic Maths groups attended seven days’ workshop on Pranayama and Vedic Mathematics respectively. Controls went jogging daily.
Assessments: Mathematics Anxiety Rating Scale, STROOP test, Children’s Cognitive Assessment Questionnaire and Digit Span Test were administered pre and post the intervention.
Analysis: SPSS-17 was used to make non-parametric pre-post comparison tests (Wilcoxon) and group comparisons tests (Mann-Whitney).
Results: Math-anxiety decreased most in Vedic Maths (-11.77±10.47; p<0.01). Others: YB (-4.08±4.99; p<0.05); JG, (-3.75±16.94). VM improved most in cognitive flexibility and reaction to cognitive stress (+9.77±5; p<0.001); YB (+5.38±5.38; p<0.01) and JG (+8.58±9.91; p<0.05). Changes in self-defeating cognition scores associated with test anxiety and performance were YB (-1.77±1.83; p<0.01); VM (-1.38±3.2), JG (+0.67±1.44). Changes in digit span scores were similar in all three groups.
Discussion: The first two groups showed improvement in cognitive skills and decreased math-anxiety compared to controls. This suggests that Vedic Mathematics decreased math-anxiety by improving mathematical abilities which may have helped enhance cognitive skills. Calming effects of pranayama practice is the probable cause for Yoga Breathing group improvements.
Keywords: Vedic Maths, Math Anxiety, Pranayama, Cognitive Skills, School Children

THREE BOOKS AND A PAPER: A GEOMETRY PROJECT
Andrew Nicholas
ABSTRACT
The aim of the project was to produce a sort of miniature Vedic mathematics version of Euclid's 'Elements', covering much less ground than the 'Elements' and doing so with greater speed and ease than Euclid achieved.

This task occupies  three books; they are: 'Geometry for an Oral Tradition',  'The Circle Revelation' & 'Eight  Essays on Geometry for an Oral Tradition'. This last book will hopefully be available later in 2016, free online
The paper 'A Much-needed Innovation in Geometry' is included because it shows how a Vedic approach helped formulate the foundations of 'Geometry for an Oral Tradition'.
A difficulty in the project was that for some one and a half centuries it has been known that the foundations of Euclid's 'Elements' are flawed. Putting this right is a major contribution of the project.
Reformulation of the foundations of geometry is done in two stages. First there is the 'Geometry for an Oral Tradition' (provisional) version, which makes use of a single axiom where Euclid needed ten.
Then, in the 'Eight Essays on Geometry for an Oral Tradition', the sole 'Geometry for an Oral Tradition' axiom is proved using two principles. The result is demonstrably rock-solid foundations for geometry.

Two matters conclude the paper: (1) a statement of the foundations of geometry, and (2) Euclid's proof of one his theorems is contrasted with the far, far briefer proof of the same theorem in 'The Circle Revelation'.

NOTE The book  'Eight  Essays on Geometry for an Oral Tradition' will hopefully be available later in 2016, free online
KEYWORDS: GEOMETRY, VEDIC MATHEMATICS, FOUNDATIONS OF GEOMETRY

TEACHING CALCULUS
Kenneth Williams
Abstract:
Calculus comes under the Calana Kalanābhyām Sutra of Vedic Mathematics. Though it is a subject usually taught later in the school career, Sri Bharati Krishna Tirthaji tells us that in the Vedic system ‘Calculus comes in at a very early stage’. This paper aims to show how Calculus may be taught to quite young children. Gradients of curves can be found without ‘differentiation’ in the usual sense, and areas under curves can be found without the usual integration process. Furthermore this approach uses a similar method for both thereby unifying these two aspects of Calculus. After the idea of a limit is introduced, in a simple way, and with a little algebra and geometry, we arrive at an easy technique that gives exact gradients of curves and areas under curves. There is no need for the complex notation that is normally associated with this subject and which adds to its perceived difficulty in school (though this notation can be introduced at the teacher’s discretion).

Swami Tirtha’s Crowning Gem
Kenneth Williams
Abstract
In his book Sri Bharati Krishna Tirthaji describes a division technique which he calls the “Crowning Gem” of Vedic Mathematics, the essential feature being that the first digit of the divisor is used to provide all subsequent digits when combined appropriately with the other digits involved. He then goes on to find square and cube roots in which the first digit of the answer is used in a similar way. The main Sutra in use for this is Ūrdhva Tiryagbhyām: Vertically and Crosswise. This approach can be developed considerably, so that powers and roots of numbers and polynomial expressions can be found, and polynomial equations can be solved to any accuracy, using the Vertically and Crosswise pattern. This paper describes this pattern, initially expressed as a formula. The operation of this pattern is then demonstrated in finding powers, and then used in reverse to find roots and solutions to polynomial equations.

A Pathway for People with Perceived learning difficulties to embrace and succeed with studies in Vedic Mathematics
Vera Stevens
Introduction
Until now people who do not have a natural aptitude with the four operations in arithmetic have not had an opportunity to enter the world of Vedic Mathematics. These people include Dyslexics, certain people on the autism spectrum, children who have missed schooling in the early years due to illness, family trauma etc., and those who are primarily visual, kinaesthetic, tactile or auditory learners and for whom the standard school approach in teaching arithmetic is a failure.